Lecture Notes in
Mathematics
Edited by A. Dold, B. Eckmann and FTakens
1461
Rudolf Gorenflo
Sergio Vessella
Abel Integral Equations
Analysis and Applications
Springer-Verlag
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Authors
Rudolf Gorenflo
Fachbereich Mathematik
Freie Universitat Berlin
Arnimallee 2-6
1000 Berlin 33, Federal Republic of Germany
Sergio Vessella
Facolta di Ingegneria
Universita di Salerno
84100 Salerno, Italy
Mathematics Subject Classification A980): 45E10, 45D05, 44A15, 65R20
ISBN 3-540-53668-X Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-53668-X Springer-Verlag New York Berlin Heidelberg
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Preface
Abel's integral equation, one of the very first integral equations seriously studied,
and the corresponding integral operator (investigated by Niels Henrik Abel in 1823 and
by Liouville in 1832 as a fractional power of the operator of anti-derivation) have never
ceased to inspire mathematicians to investigate and to generalize them. Abel was led to
his equation by a problem of mechanics, the tautochrone problem. However, his equation
and slight or not so slight variants of it have in the meantime found applications in such
diverse fields (let us mention a few from outside of mathematics, arisen in our century) as
inversion of seismic travel times, stereology of spherical particles, spectroscopy of gas
discharges (more generally: " tomography" of cylindrically or spherically symmetric objects
like e.g. globular clusters of stars), and determination of the refractive index of optical
fibres. More pertinent to mathematics think of particular (inverse) problems in partial
differential equations (e.g. heat conduction, Tricomi's equation, potential theory, theory
of elasticity - we recommend here the books of Bitsadze and of Sneddon) and of special
problems in the theory of Brownian motion. Of course, these variants of Abel's original
equation comprise linear and nonlinear equations, equations of first and of second kind,
systems of equations, and the widest generalizations consist in simply retaining in the kernel
of the integral equation its integrability in the sense that this kernel is "weakly singular".
There are several good books on fractional integration and differentiation (investigating
the Abel operator and its inverse) on different theoretical levels and on Volterra integral
equations of which Abel equations are a particular type. Let us just cite those of Oldham
and Spanier, McBride, Linz, Nishimoto, and the monograph of Samko, Kilbas and Mari-
chev. In addition, there is an ever-growing literature of research results, on applications
and on numerical methods. See, e.g., the book of Craig and Brown on Inverse Problems
in Astronomy that devotes many sections to applications of Abel integral equations and
their numerical treatment, and the recent conference report edited by Nishimoto. We
neither try to be exhaustive nor to give a balanced presentation of the various questions
involved. We also do not compete with the quoted books, but rather strive for contrast.
So, what are our intentions ? And what types of readers do we have in mind ?
We want to stimulate the flow of information between at least three sorts of people.
(i) Theoretical mathematicians also interested in applications and in application-relevant
questions.
(ii) Mathematicians working in applications and in numerical analysis.
(iii) Scientists and engineers working outside of mathematics but applying mathematical
methods for modelling and evaluation.
In the past there often has been an astonishing lack of this flow of information. This
lack becomes conspicuous when one studies basic papers on stereology written
independently from each other by biologists, chemists, metallurgists, physicists, geologists, and by

IV
authors unaware of research publications of other disciplines whose methods they could
have used instead of re-inventing them.
We treat the elementary theory and describe in detail many applications in the first
part, and present the harder topics, such as ill-posedness and the behaviour of Abel
integral operators in various function spaces, on a higher level of theory later (thereby trying
to exhibit the relevance of the results, in particular of the stability estimates, for the
applications).
In two aspects we have deliberately limited our scope. One is the theory of generalized
Abel equations (they are particularly well treated in the book of Meister), the other one
concerns discretization methods for numerical treatment. For the latter we recommend
the recent comprehensive monograph of Brunner and van der Houwen. We have restricted
ourselves to a survey on discretization schemes, stressing, however, the point (in the
literature too rarely given due observance) that, in numerical evaluation of error-contaminated
measurements, because of ill-posedness it is often better to use a crude low-accuracy
method, thereby taking account of available extra information on the shape of the solution.
What are the prerequisites to read the book ? The first three chapters should be
accessible to every student of mathematics, physics or engineering after two years of study at
university, and also to mathematically inclined students of other natural sciences. For the
rest we suppose some familiarity with basic functional analysis (like theory of integration,
If and Sobolev spaces, linear operators, and Fourier transforms).
How is the book organized ? Subdivision is into Chapters, paragraphs and sections.
Thus, by 6.4.3 we mean the third section of the fourth paragraph of Chapter 6. The
theorems, lemmas and formulas are numbered within each paragraph (ignoring subdivision of
paragraphs into sections that, by the way, not always is made). Theorem 6.4.3 is the third
theorem of the fourth paragraph of Chapter 6, and F.4.3) is the third enumerated
formula of the same paragraph. Some topics that we did not want to treat comprehensively
have been delegated to Appendices to Chapters marked with capital letters, an appendix
standing on the same structural level as a paragraph. References to the literature are
made in a self-explanatory way by giving a shortened form of the names of authors and
the year of appearance.
We are indebted to the Italian Centro Nazionale delle Ricerche and to the Freie Uni-
versitat Berlin for making possible several mutual visits of the authors for work on this
book, and we highly appreciate the readiness of many individuals for discussions,
correspondence and for critically reading parts of the manuscript. In particular we mention
Prof. Carlo Pucci who also offered us excellent working conditions in the Lstituto di Ana-
lisi Globale e Applicazioni of CNR in Florence, Italy, Prof. Robert S. Anderssen, Prof.

V
Dang Dinh Ang, Prof. Gottfried Anger, Prof. Mario Bertero, Prof. S. Campi, Dr. Paul
Eggermont, Prof. Marco Longinetti, Dr. Rolando Magnanini, Prof. Erhard Meister and
Prof. Giorgio Talenti.
For the tedious work of typing the manuscripts our thanks are due to Mrs. Silvia
Heider-Kruse and Mrs. Monika Schmidt, for typing preliminary versions to Mrs. Ange-
lika Hinzmann and Mrs. Ursula Schulze. And for the important work of carefully proof
reading the whole manuscript, checking many of the calculations, and drawing figures we
are indebted to cand.math. Vera Lenz, Dipl.-Math. Andreas Pfeiffer and cand. math.
Uwe Schrader.
Rudolf Gorenflo and Sergio Vessella

roduction
Chapter 1 Basic Theory and Representation Formulas
1.1 The Abel Integral Operator
1.2 Solution Formulas
l.A Appendix: Existence and Uniqueness in L1
l.B Appendix: List of Solution Formulas
Chapter 2 Applications of Abel's Original Integral Equation:
Determination of Potentials
2.1 General Considerations
2.2 Abel's Mechanical Problem
2.3 Throwing a Stone
2.4 The Oscillating Pendulum
2.5 The Inverse Scattering Problem for a
Repelling Potential
Chapter 3 Applications of a Transformed Abel Integral Equation
3.1 Spectroscopy of Cylindrical Gas Discharges
3.1.1 Modelling by an Abel Integral Equation
3.1.2 Complications Arising in Practice
3.2 Stereology of Spherical Particles
3.2.1 The Problem
3.2.2 Formal Solutions
3.2.3 Is the Formal Solution Correct ?
3.2.4 Calculation of Moments
3.2.5 Final Remarks
3.3 Inversion of Seismic Travel Times
3.3.1 General Considerations
3.3.2 The Flat Earth Model
3.4 Refractive Index of Optical Fibres
3.A Appendix: Linear Generalized Abel Integral Equations
Chapter 4 Smoothing Properties of the Abel Operators
4.1 Continuity Properties of the Abel Operator in U Spaces
4.2 Continuity Properties of the Abel Operator in Some Spac
Fractional Order
4.2.1 Holder Continuous Spaces
4.2.2 Sobolev Fractional Spaces
4.3 Compactness of Abel Operators
4.A Appendix: Proof of a Lemma
Chapter 5 Existence and Uniqueness Theorems
5.1 The Linear Case
5.2 A Nonlinear Abel Integral Equation

VII
Chapter 6 Relations between the Abel Transform and Other Integral Transforms 95
6.1 Relations of Abel Operators with Abel Operators 95
6.2 A Brief Account on Generalizations of Abel Operators 98
6.3 Relations between Abel Operators and the Fourier Transform 100
6.4 Relations between the Abel Operator and the Mellin Transform 107
6.5 Some Relations between Abel Operators and Hankel Transforms 113
6.6 Some Relations between the Abel Operator and the Plane Radon Transform 115
6.A Appendix: Generalized Abel Equations: Survey of Literature 121
6.B Appendix: A modified Abel Transform 123
Chapter 7 Nonlinear Abel Integral Equations of Second Kind 129
7.1 Introductory Remarks 129
7.2 Linear Abel Integral Equations of Second Kind 129
7.3 Analysis-Motivated Investigations 132
7.4 Applications-Motivated Investigations:
Problem Formulations, Newton's Law of Cooling 139
7.5 Applications-Motivated Investigations:
Survey of Literature 146
7.6 A Very Brief Survey of Literature on Numerical Methods 152
Chapter 8 Illposedness and Stabilization of Linear Abel Integral
Equations of First Kind 154
8.1 General Topics in Ill-Posed Problems 154
8.2 Preliminary Discussion of the Stability of Abel's Equation 158
8.2.1 Mechanical Problem 158
8.2.2 Inversion of Seismic Travel-Times 164
8.2.3 Other Examples and Instability Properties 165
8.3 Stability Estimates for Solutions of Abel-type Integral Equations 168
8.3.1 Auxiliary Lemmas 168
8.3.2 //-bounded First Derivative of the .Solution 171
8.3.3 Lp-bounded Second Derivative of the Solution 176
8.3.4 Discrete Data 179
Chapter 9 On Numerical Treatment of First Kind Abel Integral Equations 182
9.1 General Considerations 182
9.2 Quadrature Methods 184
9.3 Evaluation of Measurements 185
9.4 A Numerical Case Study 191
References 195
Subject Index
210

Introduction
In 1823 N.H. Abel considered the following problem of mechanics:
In the vertical (x,y)-plane (see Eig. 1) find a curve C that is
the graph of an increasing function x = <j> (y) , y £ [o,H], along which
under constant downward acceleration g a particle must be constrained
to fall, in order that its falling time equals a prescribed function
t(y) of the initial height y.
^- x
Fig. 1
In absence of friction the problem is reduced to that of solving
the equation
A)
■1/2
J (y - z) "" u(z)dz = /2g t(y), y £ [o,H]
where
u(z) = /l + <j)'2(z) .
Abel solved this equation in [Ab, 1823] and in [Ab, 1826 ] .Infact,he
treated a more general equation, replacing

2
(y - z) 1/2 by (y - z)a 1, 0 < a < 1.
For these reasons an equation of the form
B) Jau : = — J (x - t)a~1 u(t)dt = f(x), 0 < a < 1,
r (a) 0
where r is the Euler gamma function, is called an Abel integral equation.
equation B) is one of the first integral equations ever treated
(for a history of integral equations the reader may consult [Di, 1981,
p. 91 - 96], [La, 1977], [Wo, 1965]). In honour of V. Volterra (towards
the end of the 19th century) equations of type B) or of the more general
type
C) (Au) (x) := J K<x't} u(t), dt = £(x) r 0<a < 1,
a 0 (x - t)
are often called "iingulal Volttina tquatloni, o& {,ll6t kind".
We refer to an equation of type C) as an A6a£ inte.gia.1 equation
and, more generally, integral equations having a kernel with a
singularity of the type (x - t) a or (t - x) a , where 0 < a < 1 , are
also called Abel integral equations. Such a singularity is often called
a we.ak &A.n,gula>iA.£y.
The Interest for Abel operators u -» A u and Abel equations is
essentially motivated by the following five questions:
(A) Physical problems that lead to an Abel integral equation.
(B) Relations of Abel operators, in particular of the operators J ,
to other integral operators.
(C) Properties of continuity and compactness of A on certain special
functiohB spaces (Lp, C°, Ca, Wk,p,...).
(D) Questions of uniqueness and existence.
(E) Ill-posedness, in the sense of Hadamard, and numerical treatment
of the equations.
(A) Abel integral equations (not only of type C) but also more
general ones, linear or nonlinear,of first kind or second kind) have
applications in many fields of physics and experimental sciences: problems
in mechanics, scattering theory, spectroscopy, stereology,
seismology, elasticity theory, plasma physics often lead to such equations.
We discuss some of these in Chapters 2,3 and 7.
We divide these applications into four groups. The first group
(mechanics, scattering) contains problems leading to equations of
type A) and is treated in Chapter 2. The second group (spectroscopy,
stereology, seismology, plasma physics) consists of problems leading

3
to equations of the type
R k ...
D) J £ u(t) dt = f(x), k = o or k = 1,
x /72
/t - x
2
and is treated in Chapter 3. In the third group (the
radiation-diffusion problem) there are the problems leading to a nonlinear Abel
equation (of second kind). We consider these in §§s 4 and 5 of Chapter 7.
The fourth group comprises problems that can be formulated via
"ge.ne.Kalize.cL Abel eqaatloni", e.g. problems in elasticity
theory and in partial differential equations (Tricomi equation). We
briefly consider these in Appendix 3.A of Chapter 3.
For every problem we indicate its physical (sometimes also its
historical) origin, and illustrate the applications of physical laws. We
describe the mathematical approximations used and the main hypotheses
on the data and the solutions required in the formulation of the
equations .
(B) The Abel operator J of B) is a simple and useful example of
a fractional power of an operator : in particular, if a=1, then J =J is
the operator of integration, x
given by Ju(x) = / u(t)dt. The reader can easily verify (see
o
also § 1.1) that after replacement of a £ @,1) by a positive integer
J u is nothing but the a-fold repeated integral of u. These and other
precise properties justify the term fractional integral operator used
for J if a £ @,1), and fractional derivative operator for the inverse
of J :
E) Dau(x) = -A. J1"au (x) .
Many authors have worked out the idea of fractional integrals
and derivatives and have considered operators of type J and D and
ex * ex d ex *
of similar types (e.g. (J ) , the adjoint of J , -5— (J ) etc.) .
We recall Abel(l826), Liouville A832), Riemann, Weyl A917), Hardy and
Littlewood A928), Love and Young A958), Bosanquet A930, 1969), Tamar-
kin A930), Tonelli A928), Erdelyi A940, 1965, 1972), Kober A940),
Osier A970), Gelfand and Shilov A964). We refer the interested reader
to the books [01-Sp, 1974], [Ro, 1974], [McB 1979]
for an extensive literature on the subject. One of our principal
interests is a study of the relations between operators Ja , J ,
u and D , for example
dV = Da+B or DajB = JB"a .

4
Such properties can be considered as relations of the set of
operators J with itself. We treat them in § 1 of Chapter 6. Other remarkable
and useful relations can be found between Abel, Fourier, Mellin, Hankel
and Radon transforms (see §3, §4, §5, §6,
respectively, of Chapter 6). In particular the relations of the Abel operator
with the Fourier and the Mellin transforms permit us to find some existence
2
results and stability estimates in L for the equations
Fi) y—- J (x - tH-1 u(t)dt = f (x) , x e m ,
Fii) r|—r { (X - t)a_1 u(t)dt = f(x), x > 0 ,
^ ' 0
-a x
Fiii) |~-j- J (x - t)a 1 u(t)dt = f(x), x > 0.
In particular the equation Fii) is a simple example of a Wiener-Hopf
equation (see [Wi-Ho, 1937] and [Ta, 1973]). Furthermore, the operator
in Fiii) is a particular example of a fractional integral operator
studied by Erdelyi (see [Er, 1940], [Ko, 1940]).
In our study of relations with the Hankel transform we only consider
the operators
G1) A.u(X) = -^- J ^M^L. , x > 0,
1 /7 0 /x2 - t2
+ oo
Gii) A0u(x) = J- J ^M±= , x > 0.
2
r&
Concerning the relations between Abel and Radon transforms, we
restrict our attention to the plane Radon transform. We only discuss
papers of Cormack A963, 1964, 1982) who discovered and applied them.
In appendix A of Chapter 6 we briefly indicate some approaches
to a solution of the generalized Abel equation
(8) -fg} j <x - t)a~1 U(t)dt + Pf^- ; <t - x)a_1 U(t)dt = f<x)
0 < x < a,
where a £ @,1), and ¢,^, f are known. In these approaches some
relations with other integral transforms are involved, in particular with
the Hilbert transform (see [Pe, 1968], [Pe, 1969] and [Sa 1, 1967].
(C) The Abel operators J and A have several continuity and
compactness properties on important function spaces. We call them
"imootk-ing pn.oplKt-in&" . In fact, in a naive language we can say that a
transformed function J u or A u is 'Smoother" than u in terms of the
a

5
order p of summability (if we work in L^-spaces) , the order of fractional
differentiability in fractional Sobolev spaces iv '^ or the order of Holder continuity in
C spaces. The compactness of J and A (with respect to appropriate
function spaces) is a consequence of these smoothing properties. Many
authors have contributed to the discovery of smoothing properties. We
mention in particular Hardy and Littlewood who proved, for the first time,
the inequality (see Theorem 4.1.3. of § 1 of Chapter 4).
(9) ||Jau||_j_ < c(a,p)||u||
L1-ap (Q/a) Lp @,a)
1
valid for 0 < a < +«> and 1 < p < —, with c(a,p) being a constant. Many
generalizations and different proofs are known for (9), see the
references in § 1 of Chapter 4. Hardy and Littlewood have also found many other
inequalities and smoothing properties in spaces of Holder continuous
functions (see § 2.1 in Chapter 4). Many of these anticipate similar
properties of Riesz potentials (see [St, 1970]). Smoothing properties
of another type are proved for spaces w '^@,a), 0 < k < 1,
1 < p < +«>. These are also considered in more recent papers; we recall
[Kb, 1974], [Bi, 1983], [Bi, 1984]. We restrict our attention to the
case p = 1 not considered in the aforementioned papers and use the
results in the proof of existence of J u = f in L @,a) (see below).
(D) In his papers of 1823 and 1826 Abel proved that the equation
J u = f
is solved by
A0) u(x) = Daf(x) := ^- J1 f(x).
In essence he found A0) for analytic functions by applying the definition
and certain simple properties of Euler's r-function (see § 1 of
Chapter 1). In reality the representation formula A0) is valid fora class of
functions considerably larger than that considered by Abel. For example if f is
absolutely continuous then u is in L (see [To, 1928] or Theorem 1.2.1).
It is rather simple to prove A0) , and ,as a consequence, uniqueness and
existence theorems for J u = f in many function spaces, using the
smoothing properties proved in Chapter 4.
We shall consider problems of uniqueness and existence for Abel equations
in many places (Chapters 1,5,6,7), however the most systematic treatment
is to be founa in Chapters 5 and 7.
In Chapter 5 we study mainly the linear cases J u = f, A u = f
and the nonlinear equation

6
(in —L_ ? K(x't'u(t)) dt = f(x) .
A1) r(a) iQ ^ _ tI_a f(x)
For the equation A u = f we substantially refine the results of G. Ko-
walewski A930).
In §§ 2 and 3 of Chapter 7 we study the nonlinear Abel integral
equation of second kind
A2) u(x) = g(x) + T^-y J (x - t )a~1 f(x,t,u(t))dt.
o
We discuss the existence and uniqueness results of Reinermann and Stall-
bohm A971) who used in their long proof a fixed point theorem of Edel-
stein A962) .
(E) The problem of inverting the Abel operator in Lp@,1) spaces,
1 < p < +oo, is an ill-posed problem. In fact the Abel operator J :
Lp(o,a) -> l.P @,a) is a compact operator (§ 3 of Chapter 4) , hence (J )
cannot be continuous. A very important consequence is that in physical
applications of Abel equations a small error in the measured data implies
an incontrollable error in the solution. In other words, the formal
representation formulas are of dubious value in the computation of the
solution from data contaminated by noise. Indeed, for this computation
some a priori informations are required.
For general introduction to problems of ill-posedness we recommend
[La, 1967], [La-Ro-Si, 1986], [Ti-Ar, 1977], [Ta, 1978]. Here we want
to concentrate our interest to the inversion of the Abel operator. Abel's
integral equation often occurs in applications and is a simple example
of an ill-posed problem. A crucial point in the study of an ill-posed
problem consists in finding itab-Lt-ity e,&£-lma£e,& with respect to the data
when an a priori bound on the solution is known. These estimates allow
one to control the error of the solution in terms of the error in the
da'ta .
In the case of Abel's equation J u = f in [ 0, 1 ] we obtain a stability
estimate of the form
a a 1
Hull p < C(a,p) f E1+a + ||f || 1;a } !|f|| 1+a
LP @,1) LP@,1) LP@,1)
where E is an upper bound of || u'|j p, ., c is a constant and
1 < p < +o° (compare Theorem 8. 3 . 1 .) .
We study the ill-posedness of Abel equations (only in the linear
case) in §§ 3 and 4 in Appendix B of Chapter 6 and, systematically, in

7
Chapter 8. In Chapter 8 we also illustrate the physical meaning of the
a priori bound
A3) || u'|lLP@f1) < E .
We show that in many problems this bound is an appropriate and natural
condition on the solution.
Other estimates with weaker or stronger -a priori"assumptions
on the solution are presented in Chapter 8. Among these we consider the
case in which J u(x) is known, with error, only on a finite set of points
and u satisfies an a priori bound on one of its derivatives.
In Chapter 9 we survey numerical methods for first kind Abel
integral equations Au = f,thereby stressing the importance of considering
the quality of the data f with respect to their precision. The recent
monograph of Brunner and van der Houwen A986) contains detailed
descriptions of discretization methods and many references for the case
of smooth right-hand side f. Thus we need not dwell on high accuracy
mernoas. On the other hand, we
concentrate on heuristically motivated data fitting schemes for the case
of perturbed right-hand side f. These schemes use optimization methods
for incorporating shape constraints (like nonnegativity, monotonicity,
convexity) the solution u is a priori known to satisfy, and they
exhibit a rather good performance even in case of severely perturbed data
(given e.g. by a measuring device). Let us mention the papers of Larkin
A969), Eckhardt A974) and [Go, 1979 ] . The chapter concludes with the
presentation of a numerical case study carried out by W. Zikoll A981).
As a guide for the reader,we indicate below the principal
connection between the chapters of the book.
9

Chapter 1: Basic Theory and Representation
Formulas
1.1. The Abel Integral Operator
The aim of this chapter is to acquaint ttie reader with
Abel operators and Abel integral equations. We shall also give basic
formulas that we shall frequently use later.
The Abel transform of a sufficiently well behaved function u is
defined as
A.1.1) j-Arr J(x-t)a~1 u(t)dt, a<x<b ,
a
where -«><a<b<«>, at @,1), and r is Euler' s gamma function ,
A.1.2) r(x) = J tx-1e-tdt, x>0 .
0
Remarks: Note that A.1.1) is actually defined for all
real a >0 in spite of the fact, that our principal interest is for CXa<1.
If a or b is finite, the symbols < in A.1.1) can often, by continuity,
ge replaced by < .
We shall denote the Abel transform A.1.1) by Ja u(x), omitting
a
tilt; subscript when there is no ambiguity, in particular if a = 0. The
operator J is called a. ^lactJional i.nte.qKa.1 ope.la.tol , the terminology being
motivated ay the following consideration. Replace a by a positive integer
n times
is the n-fold repeated integral, where
x
J v(x) = J v(t)dt, a<x<b .
a
a
Example 1.1.1: We calculate the Abe.1 tian&fiolm o& a polynomial for
the particular case a = 0. Let
n
V
p(x) = Z awx
k=0
By the linearity of J we have

9
n a x ..k,,
k=0 X la) o (x-t) a
Z ,¾. J Ux)k
k=0 F(a) J0 (x-AxI"a
x dA
z ^ xk+a } ^ dX
k=0 F(a) " J0 A-AI"a
Now, by the well-known formula (see [Abramowitz-Stegun, p. 258,
formula 6.2.1])
A.1.3)
1 AB
J —*-
0 A-A)
— dX - TT{,l + ])Tla) for 8 > - 1
1-a T(8+1+a)
It follows that
A.1.4)
n a, r k+1
,a . , ^ k k+a
J p(x) = Z ttt;—; x
k=0
T(k+1+a)
In the particular case a = 1 we obtain the polynomial
\
. n a,x'
j'p(x) = Z ^
k=0 *+l
k+1
a primitive of p. Observe that if a * -oo the Abel operator A.1.1'
yields
n (k) , >
,a , . _ d (a) . ,k+a
J P x = z f ».,„, i x-a
k=0
T(k+a+1)
Example 1.1.2: We calculate the A6e£ tuani^oum of, a ckaiactci^At-Lc
function. Let [c,d] be an interval of IR and let Xr_ ji be the
characteristic function of [c,d], = 1 f or x £ [c ,d] ,0 elsswherav If [c ,d]<=[a ,b] a
simple calculation yields (see fig. 1.1.1)
J X
1
[c,d](x) rA+a)
(x~c) X[Cfb](x)-(x-d) X[dfb]
(x)

10
Fig. 1.1.1
We here have an example of a discontinuous function transformed via
A.1.1) into a continuous one. The converse, as we shall see later
(Theorem 4.1.4) is impossible. Of course, there exist discontinuous functions
whose Abel transforms are also discontinuous.
Example 1.1.3: Consider, for -1<A<-a the function u given by
u(x) = 0 for 0 < x < x , u(x) = (x - x ) for x < x < 1
->
Fig. 1.1.2

11
We have
~0, 0 < x < x
-- o
Jau(x)
r A4-A) , _ ,A+a
rA+A+a) (X XoJ ' x
< x < 1.
o
Remark concerning formula A.1.3): This formula is a particular case
of the general formula
1
/A-A)r AS_1 dA = r<^(s) = B(r/S)/ r>0, s>0 ,
0 l '
for Euler's beta function. By easy changes of variables we obtain two
formulas that Will be freely used in the sequel.
J (x-t)r~1 ts~1 dt = xr + s-1 B(r,s), r>0, s > 0, x>0 ,
J (q-t)r 1(t-p) r dt = / A-A)r 1 A r dA = r(r)rA-r),r>0,-»<p<q<«.
1.2. Solution Formulas
We are going to derive some representation formulas for solutions
of Abel integral equations arising in problems of physics. Although we
find these formulas by formal, arguments, they are valid for a large class
of functions. In Chapters 4 and 5 questions of uniqueness and existence
will be discussed on a higher theoretical level and in more detail.
Consider, for a£ @, 1) , -«> < a < b < «> the classical Abel equation
x
A.2.1) jT-J / (x-t)a 1 u(t)dt = f(x), a<x<b
Remark: We remind the reader that in the domain a < x < b of validity
in case of a or b being finite,the corresponding sign < sometimes may,
by continuity, be replaced by < .
. _ . -a
In order to solve A.2.1) we multiply both sides by r t T-a\~
where a<x<y<b. Integration over (a,y) yields

12
r(a)rci-a) 1 |(y_x) a / (x-t)a 1 u(t)dt| dx = rci-a) f (y-x) °'f(x)dx-
a a d.
Interchanging the order of integration on the left gives
(a)r1A_a) /{/ (y-x)"a(x-t)a 1 dx} u(t)dt = rA1_a) J (y-x) a f(x)dx.
Now by substituting x = t + AA-t) and usingA .1.3) we get
y 1
A.2.2) J(y-x)~a(x-t)a~1 dx = / \~a A-A)a" dX .
t 0
Using this formula we have
y y
A.2.3) J u(t)dt = ,] J (y-x)"af(x)dx .
a a
If the right-hand side of A.2.3) is differentiable we obtain the
formula
1 H x
A.2.4) u(x) = ' ^ J (x-t) a f(t)dt, a<x<b .
a
At this point the following questions arise:
i-i) In what clcU6U of funct-ioni -ii the. equat-ion A.2.1) un-iquely
solvable ?
l-i-i) Undei what hypotheiei on f and -in what t,enbe do ei A.2.4) hold ?
In othen. won.di,:li, the iolut-ion of (/.2./) neally g-iven by (/.2.4) ?
An answer to these questions can be found in the paper of Tonelli
[ 1923] . He proved
Theorem 1.2.1: Suppoie a,b £ IR . Then theie ex-iiti at mo&t one iolu-
t-ion of equ.at-ion A.2.1) -in L (a,b). Hotieovei, -if the fu.nctt.on f -ii ab-
iolutely cont-inaoai on [a,b] then equat-ion A.2.1) hai a iolut-ion -in
1
L (a,b) , graven by foK.mu.la A.2.4).
As an application of Theorem 1.2.1 and formula A.2.4) we solve the
mechanical problem (see formula A) of the Introduction) for two
particular choices of the function t(y), namely
(a) for t(y) = const, the -iioch>ione pnoblem,
(b) t(y) = -/ly/q,falling time as in ^nee {,0.11.

13
(a): The. ca&e. t(y) s c = const. Recall that in the mechanical
problem the equation is
A.2.5)
J /l+*'2(g) d? = t(y) .
0 /2g(y-C)
In the isochrone problem,t(y) = c = constant. Now by formula A.2.4)
we have
Z ,2 . , 1 d Y( /2g c
n dy
dt , ip@) = 0 ,
that is
A.2.6)
0 /y-t
i+ip'2(y) = -^- 1 , cp(o) = o .
It is well known that the graph of the solution x = ip(y) of A.2.6) is
an are of a cycloid that has the parametric representation
2 2
y = 2£_ (i-cos t), x = 2£_ (T + sin T), o < t < n ,
or, in explicit form,
<P(y) = —p arccos ( 1-
gc '
2gc 2
-X~2 y-y
2gc
o<y < -^-
Fig. 1.2.1

14
(b) : The. cai>n t(y) = /2y/g . By A.2.4) we have
Y
/w2(y) = | £ / -^= d? , <p
Formula A.1.3) then yields
/l+ip'2(y) = 1, ip(O) = 0 ,
@)
that is cp ■ 0, as to be expected (see Fig. 1.2.2)
A
H ■
i
Pig. 1.2.2
In analogy with the fractional integral operator, the operator
x
u
1
; "(t)dt = d jl-a = a -1U
dx r A - a) J , ,a dx a
a ix tj
is called a fractional derivative operator and is denoted by -r— J
(J°) , D - We omit the subscript a when there is no ambiguity.
As an example,we calculate the fractional derivative of a
polynomial
p(x) = Z ak.x
for the special case a = 0. We have, see A.1.1),
d 1-a , . _ d
dx J plxJ dx ^ r(k+2-a
n ^r(k+i) k+1_a
£ Trn—^ . . x
" \r<k+1> k-a
k=0
r(k+1-a)

15
Therefore
n a T(k+1)
°P(X) " * HkTT^T* ■
k=o
We observe that if we formally pose a = 1 in the last formula we do not
generally have D p(x) = -=— p(x) . In fact, the first term
a TA)
o -a , . . c
r ,. _—r- x has no meaning if a = 1 .
The formula A.2.4) can be written in different forms. In fact, an
integration by parts gives (we suppose a £ IR )
J (x-t)"a f(t)dt = Jll*)(x-aI-a+ _L ) (X_tI-af.(t)dt .
a a
Therefore the solution u of A.2.1) can be written as
A.2.7) u(x) = r(i_a) {f(a)(x-a)"a + J (x-t)"a f'(t)dtj .
This formula suggests writing u alternatively as a Stieltjes
integral (see appendix 1-A) . If -co <a and f is extended by putting f(x) = 0
for x<a we have
A.2.8) u(x) = rHl , J (x-t)~a df(t) .
K ' a-0
If a = -co and Ixl f (x) -»0 as x -> - <» we have
A.2.9) u(x) = T,]-a) / (x-t)~a f (t)dt - r(T_a) J (x-t)~adf(t)
—oo —oo
From the representation formulas for the solution of A.2.1) we can
find,by an easy change of variables, the solution of several other types
of Abel integral equations.
Consider the Abel equation
b
A.2.10) YTT / (t-x)a u(t)dt = f (x) , a <x < b .
x
Substituting £ = b + a - t we get
b+a-x ,
A.2.11.) jrj^j J (b+a-C-x)a u(b+a-C)d? = f (x) ,
a
a < x < b .

16
A second substitution X = b+a-x, U(£) = u(a+b-£) yields the
equation
A.2.12) yr]—j- J (X-Qa~1 U(C)d? = f(b + a-X) , a<X<b ,
which can be solved by the representation formula A.2.4). By inverting
the substitution we find
1 a b
A.2.13) u(x) = T{\_a) ^ J (t-x) af(t)dt, a<x<b.
In many physical applications (see Chapters 2 and 3) there arise Abel
equations in even more general forms:
A.2.14)
1 ? r„, , .,^,0.-1
r
^- J [h(x)-h(t)]u u(t)dt = f(x), a<x<b ,
1 b -1
A.2.15) J— J [h(t)-h(x)]a u(t)dt = f(x), a<x<b .
X
where h is a strictly increasing differentiable function in (a,b). Espe-
2 1
cially important are h(x) = x , a = 0, a = -^, see Chapter 3 .
We treat the equation A.2.14) by the substitutions £ = h(x),
t = h(t), a1 = h(a), b' = h(b) and then put
v(t) = u(h"\(T)) , g(C) = f(h-1(e)).
h'(h (t))
We obtain
A.2.16) 1 J v(T)dT - g(g), a'<g<b- ,
vaj a' (£-t)
hence with the representation formula A.2.4) and by resubstitutions
, x
A.2.17) u(x) = ^ r(i-a) J [h(x)-h(t)]a h'(t)f(t)dt, a<x<b .
Analogously we obtain for A.2.1s)
_J
'A-a)
, b
A.2.18) u(x) = - ■— rA.a) J [h(t)-h(x) ] a h' (t)f (t)dt,
a < x < b.
Another method for solving A.2.14) (and analogously A.2.15)) was
given by [Srivastava, 1963] (see also [Burlak, 1964] and [Sneddon,1966])

17
Consider the equation A.2.14). Multiplying both sides by
1 h' (x)
r A-a) r, . , ,, , -.a
[h(y)-h(x)]
with y£ [x,b] fixed, we obtain, in analogy to A.2.1), the equation
1 X f X
r(a)rA-a)
a *■ t [h(y)-h(
' (x) dx I
x)]a[h(x)-h(t)]1-a J
/ u(t) J 2-i£i S* __ dt =
Y( h' (x)f (x) dx
a [h(y)-h(x)]a
The inner integral on the left hand side of this formula is,
by A.1.3), equal to r (a) r A -a). Hence A.2.17) is valid.
1
Appendix 1.A : Existence and Uniqueness in L
Here we prove Theorem 1.2.1 in the following more general form.
Theorem 1 .A. 1 : Suppoie a,beiR, a<b. Then theie exlhtt, at mo&t
one solution o$ equation A.2.1) -in L (a,b). Hoieovei, If the function
f li of bounded va/ilatlon and contlnuoui fnom the light then equa-
tlon A.2.1) hai a solution In L (a,b) , given by
A.A.1) u(x) =-1 J ^L
rA"a) a-0 (x-t)a
wheie the Integral It, In the Lebeigue-$tl€&tjei ienie.
Proof: (a) for uniqueness, (b) for existence .
1
(a) Let u£L (a,b) be a solution of
A.A.2) _L_ x u(t)dt _ a<x<b
Ha) J ~ ~T^ ~ ° ' a<x<b .
a (x-t)
Consider, for a fixed y£ (a,b), the function
u(t)
A.A.3) (x,t) «
(x-t) (y-x)

18
defined in the triangle T = { (x , t) | a < t < x < y}
A
-> x
Fig. 1.A.1
1
The function A.A.3) is in L (T ) , since by Tonelli's theorem (see
[Royden, 1968]) we have
//
u(t)
T ' (x-tI a(y-x)a
Yr Y
dt dx = ; iu(t)i ;
a ^
dx
t (y-x) (x-t)
dt
= r(a)r A-a) J u(t) | dt < + «. .
a
Now by Fubini's theorem (see [Royden, 1968]) and A.A.2) we have, for
every y £ (a,b) ,
/ u(t)dt = ; {u(t) r(a);A_a) /
dx
t (y-x) (x-t)
dt
Yr \_\ 7 "(t)dt 1 dx =
' ]T(a) J . M1-a J rM ., , a
a l o (x-t) ' TA-a)(y-x)
= 0
Therefore
u (y) = 0 for a < y < b
(b) Let f be as assumed. Then there exist two functions i.,£-,
increasing, of bounded variation, continuous from the right, such that

19
f = f. - f_ and hence df = df. - df., . Therefore if u is given by
A.A.1), that is
1 ( x df 1 (t) x df (t) -,
<1-A-4) U<X) = ^T^T { I 7^ - S-~—-a \ ■
La-0 (x-t) a-0 (x-t) '
we have
1 x df^t) x df2(t)
l^(t) I < mr^- J ——— + J ——— .
a-0 (x-t) a-0 (x-t)
To show that u£L (a,b) it suffices to prove that
„ ^ j dtp(t)
a-0 (x-t)a
is in L (a,b) for any function ip increasing, bounded, continuous from
the right on [a,b], and extended by 0 on the left of a. we have
j j M^dt = J ;^_^dip(C)= i j (b-o1"^^?)
a-0 a-0 (t-C)a a-0 £ (t-?H a-0
., . 1 -a b ,, ,1-a
(b-a) f , . ... (b-a) ,, ,
< i_n J dip(C) < ' ' ip(b) < + «. .
u a-0 " ' a
Hence u[L (a ,b) .
Now the function
(t,C) ~ —
(x-t) (t-C)
is in L (T ,dt 8 dtp(g)) for a<x<b and any ip having the aforementioned
properties.
In fact, by Tonelli's theorem,
]_ X ( dt 1 ) dtp(g) 1
F(a) a WtI~a rA-a) a-0 (t-C)a J
= r(a)rA-a) J dtp(S) J t-a = tp(x) '
i va;i u a; a_Q ? (x_fc) i a(t_?)a
Now use A.A.4) to conclude the proof.
Observe that if f is absolutely continuous in [a,b] then df(x)
= (f(a) 5 (xTa)+ f'(x)) dx , where S is Dirac's "delta function". There-

20
fore Theorem 1.2.1 and formula A.2.7) are consequences of Theorem 1.A.1
and formula A.A.1).
It is useful to observe that the formula A.2.4) is more general
than formulas A.2.7) and A.A.1). In fact, consider the function
0 , 0 < x < x
f(x) = ' r A + A) , ,\+a ..
'- (x-x ) , x < x < 1 ,
rA+A+a)
where x c @,1) and -1 <A< -a . This function is not of bounded variation
in [0,1], hence formula A.A.1) cannot be used, but, as we have seen in
Example 1.1.3 the function
u(x)
, 0 < x < xQ
(x-x ) , x < x < 1
o o
is solution of
A A 5) ^- f u(t)dt = f(x)
1 Ha) J . „.1-a t(x)
Furthermore u is in L @,1) and is therefore the unique L -solution
of A.A.5). We also observe that
. 0<x<x
J x £(t)dt '
TA-a) 0 (x-t)a (x-x )A+1
o'
X < X < 1
1+\ ' o
Hence x v* _ ,._—r- / —-—' is absolutely continuous and
U a> 0 (x-t)a
J 1 jiitidt =u(x).
dx rA"a) 6 (x-t)a
The true reason for the difference between the formulasA.2.4) and
A.A.1) or A.2.3) lies in the fact that the set of functions of bounded
variation in [a,b] is a proper subset of Ja(L1 (a,b)) for any a e @,1)which
should be clear from Theorem 1.A.1 and Example 1.1.3.
In Chapters 4 and 5 we shall give a more precise characterization
of J (L (a,b)). At present we conclude our considerations with the proof
of the following theorem.

21
Theorem 1.A.2 . Let - °° < a < b < + °° . Then theie exliti a ^unct-lon
u C L (a,b) iuch that
A.A.6)
1 ( u(t)dt
T^T a (x-tI'0
= f (x) , a < x < b ,
-CE and only -ifa f £ L (a,b) and .the function
A.A.7)
r1-a
J f(x' = rTwo J
1 f f(t)dt
(■1-a) J / t,o
a (x-t)
1-a,
, a < x < b ,
■Li abiolu.te.ly cont-Lnuoui with J f(a) = 0.
Proof: If there exists u€L (a,b) satisfying A.A.6) then
b x ,.. ,. , , ,^. , b , b
b
J I
a
¢/ ^^ 1 r ^ r lu(t)|dt 1 f f. ,,.,, f dx \ ,.
f (x) idx _< ^ ; ax ; --^ = ^ j |iu(t) i j —-^ } dt
1 ? (b-t)a
(b-a)
a b
r(a) { a
|u(t) Idt < rn + 'a) J lu(t) Idt ,
- T 1
hence f cL (a,b). Furthermore
1
J
f (t)dt
1
dt
x (• x
rA"a> a (x-tH' " TTT^aTTW ^ 1U(S) { 7x-t)«(t-C)
YT^m = / u(?)d?
We have proved the "only if" part.
To prove the "if" part assume that the function in
A.A.7) vanishes at x = a and define
u(x)
1 _d_ *j £(t)dt
rA-a) dx ' . ..a
a (x-t)
Then u £ L (a,b) and
x
A.A.3) J u(S)dS -^if^
a a (£-t)
C=x
f (t)dt
Now
rA-a) ' . ., a
C=a a (x-t)
X ,f. ,f 1 x dt / 1 ) u(g)dg \
/ u(?)d? = , J I p-^y J i_a j .
a l { ' a) a (x-t)a \ * (a) a (t-C) a '

22
Therefore, by A.A.8),
_j x ;_l_ J ^(g)dg f (t)i at =
Fd-a) a tr(a) J (t_?I-a f(t)j (x_fc)a
, f iL1(a,b) and ^-J—r- J u(g)d^_ £L1(a,b) (by the proof of
the "only if" part), therefore by the uniqueness part of Theorem
1-A.1 we have:
__L_ * ^(g)dg f(x) = 0
r(a) J , e,1-a lxJ U •
a (x-C)
Appendix 1.B: List of Solution Formulas.
For the reader1s convenience we list the equations we have
considered and their solution formulas. Unless otherwise stated, we have
—=> < a < b < => .
(I) The equation
1 x n-1
A.B.1) YJ^J I <x-t) u(t)dt = f(x) , a<x<b ,
is solved by
A.B.11) u(x) = rA-a) S I (x-t)"a f(t)dt, a<x<b .
a
If a and f(a) are finite, then
CI.B.Ui) u(x) = r(]_a) { f (a) (x-a)"a + J (x-t)"a f' (t)dtj ,a <x <b.
If a is finite and f is extended by 0 to the left of a we have
1 x
d.B.Uii) u(x) = '_ J (x-t)"a df(t) , a<x<b .
( ' a-0
If a = - => and lim Ixl f (x) = 0 we have:
x -» -oo
1 x
d.B.Uv) u(x) = rA-a) J (x-t) a f'(t)dt, - °=<x<b,
—oo
1 x
d.B.lv) u(x) = rA-a) ^ (x-t)~a df(t) , -o=<x<b .

23
(II) The equation
A.B.2)
is solved by
A.B.2i) u(x)
iq-r- J (t-x)a 1 u(t)dt = f(x), a<x<b,
' X
1
rA-a) dx
d J (t-x) af(t)dt( a<x<b.
If b and f(b) are finite then
1
A.B.2ii)
u(x)
TA-a)
-- 1
f(b)(b-x) a - / (t-x) a f'(t)dt[ , a<x<b.
x '
If b is finite and f is extended by 0 to the right of b we have:
A.B.2iii) u(x)
1
TA-a)
b+0
J (t-x) a df (t) , a <x < b
1-a
If b = t » and lim |x| f(x) = o we have
X->+ oo
A-B.2iv)
u(x)
1
rA-a) dx
d j (t-x)~a f' (t)dt , a < x <
A-B.2v)
u(x)
1
rA-a) dx
d J (t-x) a df (t) , a < x < + =°
(III) The equation (here h is differentiate and strictly increasing)
x
A.B.3)
is solved by
T(a)
5I,,, "!!!??-„ = £<«> • a<x<b •
a [h(x)-h(t)]
(....31, „.,.'£ ,_>■(«»»« , a<x<„
If a and f(a) are finite then
A.B.3ii) u(x)
r(
1-a) {
a [h(x)-h(t) V
h'(a)f(a) . X h'(
/
(t)£(t)dt }, a<x
(x)-h(t) ]a >
< b.
[h(x)-h(a) ]" a [h(>
If a is finite and f is extended by 0 to the left of a we have
(....3ii» .w-nb^) ;;;';";;;.. .<.<-
a-0 [h(x) -h(t) ]

24
(IV) The equation (here h is differentiable and strictly increasing)
A.B.4) TTWT i UU) dt i_. = £<x> < a<x<b
x [h(t)-h(x)]
is solved by
A.8.41, u(x) --^jlj hMt)£(t)dt >a<x<b .
1 A a) dx x [h(t)_h(x)]01
If b and f(b) are finite then
d.B.4ii) u(x) = 1 { h'<b>£'b> „ - J h'(t)£(t)dta
1 n °° L [h(b)-h(x)]a x [h(t)-h(x)]a
If b is finite and f is extended by 0 to the right of b we obtain
A.B.4iii) uixl^-^T ^<t)f(t)^ f a<x<b .
1 A a) x [h(x)-h(t)]a
(V) Other particular cases (with a = 0)
The equation
A.B.5) J U(t)dt = f (x) , 0 <x<b ,
0 /2.2
yx -t
is solved by
A.B.51) u(x)=2dX tf(t)dt ,0<x<b ,
n dx 0 n 2
Vx -t
x f' (t)dt \ n
x | !—' >, 0 < x
A.B.5ii) u(x) = | \f@) + x J ^ v w^ ^ o <x<b
0 v/x-t
A.B.5iii) u(x) = — J d£(t) , 0 <x <b
•0 / 2 Ji.
Vx -t
The equation
A.B.6) J U(t)dt = f(x) , 0<x<b
x /~2 2
yt -x
is solved by
A.B.61) u(x) = - 1 A J tf(t)dt Q<x<b ,
n dx x rs—2
Vt -x

25
(x) = 2x ( f(b) _ ) f (t)dt 1 Q<x<b
(x) = - ix bJ° df(t) _
n x /T~2
yt -x

Chapter 2: Applications of Abel's
Original Integral Equation:
Determination of Potentials
2.1. General Considerations
This chapter is dedicated to direct applications of the integral
equation
1 x _
B.a) yj^j J (x-t)a u(t)dt = f(x),
0
Here 0 < t < x < b, a £@,1), b e JR U {«} . From Chapter 1 we know that its
solution is (formally)
t
B.b) u(t) = r(i_a) ^ J (t-x) a f(x)dx
0
= 17½ {^l + J(t-x,-f.(x,dx}
t 0
for 0 < t < b < «> or 0 < t < b = «>
Usually a = 1/2, this exponent already arising in the very first
application, namely Abel's mechanical problem.
An important problem in particle physics and physical chemistry is
that of determining potentials from scattering experiments. Particles
are shot against a target (e.g. an atomic nucleus), and from the amount
of deflection (often measured as an angle) one wants to gain information
on the potential of the attracting or repelling field of the target. This
potential depends on a radial coordinate r only. Of course, many
particles are shot against many targets, so one has to think about methods of data
collection etc.. But we leave these details to the experimenter and
restrict ourselves to the nicer mathematical aspects.
In 2.5 the classical approach assuming Newton's mechanics as valid
will be described, based on considerations of [La-Li, 1966][Ke-Ka-Sh,1956],
and the lucidly written expository paper of J.B. Keller, [Ke,1976].
See also [Br ,1976/77 ] . To readers interested in the quantum-theoretical
approach-, we recommend [Bu,1974], [Mil, 1969] , [3a1, 1972] , [ Sa2, 1973].

27
Before going into the general theory with radial dependence of the
potential,we shall present three simple model problems. The first one
is Abel's famous mechanical problem (see Introduction );the second
one is the problem of determining, by throwing a stone, a potential
depending only on one cartesian coordinate; the third problem is that of
reconstructing a potential from measurements of the duration of
oscillations of a pendulum.
Whereas in Abel's mechanical problem and in the pendulum problem
the measured quantities are durations (of times) depending on (initial)
kinetic energy, they are distances depending on (initial) kinetic
energy in the case of throwing a stone, and they are angles (of deflection)
depending on a so-called "impact parameter" (which is a distance) in
the problem of determining an attracting or repelling potential of a
point .source.
2.2. A Recent Formulation of Abel's Mechanical Problem
We present the problem investigated by [Ab,18 23J, [Ab,,18 26] , in a
picturesque formulation due to [Ke,1976], who called it "determination
of the shape of a hill from travel time". The situation is sketched in
Fig. 2.2.1 .
A
The shape y = y(x) with y@) = 0 is to be determined by throwing
at time t = 0 a particle of mass m>0 gliding without friction hill-
upwards and measuring the time at which it reaches tile ground level y=0
again. The function y = y(x) is assumed to be differentiable and
strictly increasing for 0 < x < °° . With v as initial velocity (x=0,
o j > »
y=0, t=0) the particle's initial kinetic energy is E = § vl an(j with
2 o
g as acceleration of gravity, its potential energy at the point (x,y)
i s v = mgy.
y
Fig. 2.2,1 Shape of a hill

28
Parametrizing the profile of the hill by x=x(s), y=y(s) with s
as arclength, x@) = y@) = 0, we see that on the hill, V = V(s) is
strictly increasing, V@) = 0. Thus the inverse function s = s(V) does
exist and is strictly increasing, s@) = 0.
Intuitively we expect the particle to glide upwards, until it
reaches at t = T(E)/2 a maximal altitude y = y(E), and then to glide
downwards again (if T(E)/2 is finite) returning to the origin at t = T(E).
So we have to look for an expression giving T(E) from V(s). From
the differential equation
m s"(t) = - dV(s)/ds , s@) = 0, s'@) = v >0 ,
governing the movement of the particle, s = s(t) being its position at
A
time t, we obtain by multiplication with s'(t) and subsequent
integration the conitancy of, e.ne.igy, namely
B.2.1) -j(s'(t)) + V(s(t)) = E = 2 v0 '
hence
B.2.2) t = (m/2I/2 J (E-V(a))/2 da
0
as long as V(s) < E .
If V(s) <E for all 0 < s < «> the hill is of finite height getting
flatter and flatter as one goes upwards, and t -»«> as s -»«>. In this case
the particle remains moving upwards forever and never returns.
Remember that we have introduced the function s = s(V) inverse to
V = V(s). If s(E) exists (and then is finite), we have
s (El
B.2.3) T(E)/2 = (m/2I/2 J (E-V(a))~1/2 da .
0
This value is finite if V'(s(E)) >0, but infinite if V'(s(E)) = 0 and
V"(s(E)) exists. In the latter case also the particle never returns, but
V(s) being strictly increasing, there are values E>E with V'(s(E)) >0
whence T(E)/2 finite.
A A A
Assuming E < E (where E is so chosen that for V < E the inverse
function s = s(V) exists) we get from B.2.3)
s t F)
B.2.4) T(E) = BmI/2 J (E-V@))/2 da .
0
3y a change of variables, we get the classical Abel equation.

29
With s = s(V) we obtain (equivalent to B.2.4))
B.2.5) J (E-V),/2 s'(V)dV = T(E)//Zm, for 0<E<E .
0
Solving for s' (V), using B.b) and subsequent integration yield
B.2.6) s(V) = — J (V-E),/2 T(E)dE for 0<V<E ,
n/2m 0
A A
and inverting again we get V = V(s) , 0 < s < s = s(E) .
2 2
From V(s) = m g y(s) and (x'(s)) +(y'(s)) = 1 , the following
parametric representation of the shape x(s), y(s) is obtained:
B.2.7) x(s) = J /l- (y' (a)J da , y(s) = ^T •
0 g
Remark: In nature there are many hills whose profile cannot be
described by a strictly monotonic function y = y(x). For the peculiar
difficulties facing our method of attack, we refer the reader to
[Ke,1976].
2.3. Throwing a Stone
A potential in the upper half-plane y>0 of a Cartesian (x,y)-plane
is assumed to depend on y only: V = V(y) with V@) = 0. We assume V(y)
to be strictly increasing and differentiable.
Throw a stone from x = 0, y = 0 with initial velocity components
a>0 (horizontal), b>0 (vertical) at time t = 0. Vary b, but keep a
fixed. The stone either escapes to infinity (if b is large enough) or
rises to a maximum height y* and then falls down again until it
reaches the ground level y=0 at the horizontal coordinate position x = C .
From measuring the values of £ corresponding to various values b, we can
determine the potential V from an Abel equation.
By the law of conservation of energy, we have
m, ■ 2 -2. TT/* m, 2 .2. j=„ ^ ^
p-(x +y ) + V(y) = ^(a +t> ) f°r t>° •
Since V does not depend on x,we have x(t) = a, and with the notation
8 = ^ b t we obtiain that the stone rises until V(y) = 8, at maximal
height y = y* with V(y*) = 8 (if B <sup{v(y) |y > 0} < «. ). The falling
time being equal to the time of rising, we obtain for the stone's
total flying time
B.3.1) t(8) = 2/mJT. J (8-V(y))/2 dy .
y=0

30
Now invert V = V(y) to y = y(V), take into account £(B) = a t(8)
to obtain the integral equation
B.3.2) J (B-V)~1/2 y'(V)dV = £(B)/(a /2m) for 0<8<8 ,
V=0
where 8 should be in the range of the potential function V.
Using B.fc>) we obtain y'(V) and then by integration
B.3.3) y(V) = | (V-8)/2 C(B)d8, 0<V<8
tt a /2m 0
A
Now invert again to get V = V(y) for 0 <y <y
2.4. The Oscillating Pendulum
We consider a particle with mass m>0 oscillating in a symmetric
potential well, i.e. in a potential V(x) with V(-x) = V(x), V@) =0,V(x)
strictly increasing for x>0. For convenience,let V be everywhere diffe-
rentiable. Denoting by x = x(V) the nonnegative solution of the equation
A
V = V(x) (for 0<V< sup{V(x) I x e IR } = V) a particle with total energy
B.4.1) E=^x2+V(x) ,0<E<V,
oscillates forever between x = - x(E) and x = x(E), the period of one
oscillation being T(E) = 4 T* where T* is the time that the particle
needs to travel from x = 0 to x = x(E).
From B.4.1) we obtain
x(E) ,
t* = /ST? ; dx
hence
0 /E-V(x)
x(E)
B.4.2) T(E) = /3m J dx - , 0<E<V, if T(E) < «° .
o /E-V(x)
Inserting the inverse function x = x(V),we again get the Abel integral
equation, namely
1/0 A
B.4.3) J (E-V)" ' x'(V)dV = T(E)//3m , 0<E<V ,
0
from which, by A.b) x'(V) and by integration, x(V) can be obtained:

31
B.4.4) x(V) = - -~ J (V-E) 1/2 T(E)dE, 0<V<V .
n \/8m 0
By inverting ,we get V == V(x) = V(-x).
Dropping the assumption of symmetry , we obtain the general problem of
a potential decreasing in x <0 and increasing in x >0. In this case,
there are two inverse functions, x^ and X2 with x-| (V) iO and X2<V) >0.
The reader can verify ([Ke,1976], [La-Li,1966]) that in this case it is
possible to obtain x2(V)-x1(V) by solving an Abel equation; Tiaowever it is
not possible to get these functions individually.
2.5. The Inverse Scattering Problem for a Repelling Potential
Consider a repelling center with a potential V = V(r) where r is
the distance from the center. We assume V(r) to be strictly decreasing
for r>0 and dif f erentiable, and for def initeness ,let V (°°) = 0. At r = 0
the potential may be infinite.
The movement of any particle with mass m > 0 in the field of this
repelling center is constrained to a plane in which we introduce polar
coordinates (see Fig. 2.5.1) r,tp .
Fig. 2.5.1 Particle trajectory in a repelling field
We consider a particle coming from infinity and having total energy
E = j v^ (at infinity the total energy is the kinetic energy because
V(«°) = 0). The shape of the particle's trajectory is then determined

32
by its "impact parameter" b, which is equal to the closest distance to
the center if the particle would fly in a straight line, i.e. if the
potential were everywhere equal to zero. We assume that b>0.
In the presence of the repelling field, however, the particle's
trajectory is not a straight line. The particle, as time t proceeds,
comes nearer and nearer to the center until it reaches a nearest point
with r = r , and then its distance gets larger and larger without bound.
Actually, the trajectory is a hyperbola-like curve with two asymptotes,
corresponding to t -» - => and to t -» + => , respectively, and the particle
is deflected by an angle 9 (see Fig. 2.5.1), the angle caused by these
asymptotes.
Denoting by r = r(t), ip = tp(t) the position of the particle, by
2 .
M = m r ip its angular momentum (which is like E, an invariant of the
movement) we have (compare, e.g., [La-Li,1966])
2
B.5.1) | r2 + -2L-2 + V(r) = E .
2mr
Looking at the far-away particle, we see that M = -bmv , and using
E = ^r v we can eliminate M to get
B.5.2) m r2 + b2 E r~2 + V(r) = E .
We further see that
B.5.3) ip = —y = - —-—? < 0 for all t ,
mr v'm r
so ip is steadily decreasing (an important qualitative property of the
trajectory). From B.5.2) we find
B.5.4) r(t) = ± /^~(E-b2 Er-2 - V(r)I/2 ,
m
the " - " f or - co < t < t , the" + " fort <t<«>, with r(t)=r
o o o o
The value r is characterized by
o J
B.5.5) E - b2E r~2 - V(r )=0 .
From B.5.1) we deduce
dt = \fmf2 (E-b2E r - V(r))/2 I dr I ,
and then from B.5.3)

33
, W2E ,. -2,,-2 -2 ,,-1,-2,,, ,,-1/2,, ,
dtp = - __ •) dt = - r (b - r - E b V(r)) ' I dr I ,
v'mr
so that we obtain
B.5.6) 9 = 9(b) = it - 2 J —
r 2 .,-2 -2 -1,-2,,, ,,1/2
0 r (b -r -E b V(r)) '
where r solves B.5.5) (by monotonicity r exists and is uniquely
determined).
We here write 9=0(b) because our aim is to derive an integral
equation which allows us , knowing the dependence of the deflection
angle 0 on the impact parameter b>0,to calculate the unknown poten-
tial-V(r) for r >0. The energy E is kept fixed while, b is varied.
Following Keller, Kay and Shmoys, see [Ke-Ka-Shm,1956] we introduce
new variables
B.5.7) x = b~2, u = r~1 , 9(b) = 9(x), V(r) = V(u) ,
B.5.8) 8(x) = 1 (tv - 9(x)) ,
and see that with u = 1/r the equation B.5.6) is equivalent to
u
0 ,
B.5.9) J — = 8(x) .
U=0 (xA-V(u)E-1)-u2I/2
This is not yet Abel's equation. To proceed further,we make still
another substitution
B.5.10) v(u) = 1 - V(u)/E, w = u2v, g(w) = v/2 ^ .
Note: V(r) is decreasing, V(u) increasing, v(u) decreasing but always
> 0, w(u) increasing, 0<w<x = ~ (if we restrict measurements
b ,
min
to b > b , > 0) .
- min
Using B.5.10) we immediately arrive at an Abel integral
equation
B.5.11) J g(w)dw = 8(x), 0<x<x
w=0 , ,1/2
(x-w)

34
where the square root in the denominator being zero for x = w gives
us that u = u corresponds to w = x .
From B.5,10) we obtain, using B.b),
/->ri-,* /* 1 d r 6 (x) dx
B.5.12) g(w) =-^/ l ' y2 ■
x=0 (w-x)
How to calculate V(r) from g(w)? B.5.10) gives u = \/vw, whence
1 -1 1/2 dv 1 -1/2
g(w) = 1 v w _ + _ w / .
From this differential relation, we obtain the dependence of v on w:
w ---1/0--1-
B.5.13) v = v(w) = exp JBg(w)w ' - w )dw.
0
Here we have used the facts that u = 0 implies w = 0 and that
V@) = V(°°) =0 implies v@) = 1.
What have we obtained ? A parametric representation of the
dependence of V on r, namely (see again B.5.9), B.5.10) and use r = 1/u)
B.5.14)
Remark: It should be noted that if the repelling field is very
strong near r=0, then even ' when varying the impact parameter b in the
whole range from 0 to » for a fixed energy E > 0, there may be a minimal
distance r* >0 such that the method gives no information on V(r) for
r<r*, but only for r > r* . The reason is that r* = inf{r b > 0} may
be positive. To obtain information for still smaller values of r,one
must then take a greater value of the energy E in order to come nearer
to the source of the field.
We shall not discuss in detail these and related problems.
We refer to the specialized physical literature for the problem
of identifying the dependence of the deflection angle 8 on the impact
parameter b if very many particles are in a parallel beam shot against
many repelling centers.

Chapter 3: Applications of a Transformed Abel
Integral Equation
We consider here several problems leading to Abel's integral
equation in the form
C.a) J u(t)dt = f(x) .
x /~2 2
Vt -x
Here 0<x<t<b<«> . Appropriate solution formulas are
,-,,, ,., 2 d r x f (x) dx
C.b) u(t) =--^/
J^?
t ./72 ^2
b f ' (x)dx
C.c) u(t) = 2t (_f(bI_ ;
" lv^V fc v£^2
t
The applications described have all come up in the twentieth
century, and we cannot be complete in listing all such applications. We do
not adhere to chronological order, but we begin with that case where
the modelling process is most direct.
3.1. Spectroscopy of Cylindrical Gas Discharges
3.1.1. Modelling by an Abel Integral Equation
Consider a gas discharge in a cylindrical tube (with circular cross-
sections) of radius R emitting radiation whose intensity g(r) is assumed
to depend only on the distance r from the central axis (see Fig. 3.1.1).
One wants to determine g(r) for 0<r<R by measurements from outside.

36
5> x
line of measurement
Fig. 3.1.1 Discharge tube cross section
Hormann, see [Ho,1935], seems to have been the first to use Abel's
integral equation for modelling the configuration of "side-on"
measurements, and in recent decades through the growing importance of
plasma physics many research papers have been devoted to this topic ; most
of these deal with various numerical (computer) procedures and their
performance. For reviews see [Go,1979] and [Br ,1932] and Chapter 9.
It is of interest to note that exactly the same modelling is suitable for
investigating radially symmetric objects in the sky, e.g. globular
clusters of stars with respect to their smeared-out density (see[Bra,1956])
and[Cra-Br,1936]).
Returning to our problem of finding the intensity g(r) for 0<r<R,
we imagine a detector moving outside the plasma parallel to the x-axis
(see Figure 3.1.1) which measures the integral
y(x)
C.1.1) G(x) = J _ g(r)dy, 0<x<R ,
y=-y(x)
of the intensity g along parallels to the y-axis in a plane
perpendicular to the axis of the cylinder. Taking account of
J = J and of r = x + y ,
-7 0
x2 + (y(x)J

37
(Pythagoras' theorem) we obtain the IntlQial equation
C.1.2) 2 J 9(r) r dr = G(x) , 0 <x < r < R ,
r=x /*2 2
yr -x
for determining the unknown function g(r) from the known function G(x) .
By C.b) the solution is
C.1.3) g(r) = - -L * / G(x_)x.dx
nr dr
*- yC
2-r2
Remark: Under the natural assumption that in a vicinity of the
boundary r = R the intensity g(r) is bounded, lg(r) I < K< °°, we deduce
from C.1.2) that
IG(x) I < 2 K /r2-x2
in a vicinity of x = R and in particular G(R) = 0 .
By the alternative solution formula C.c) we find
R
C.1.4) g(r) = -1/
G' (x)dx
I r
x = r
dG (x)
./2 2
/~2 2
\/x -r vx -l
at least if G is differentiable or if the Stieltjes integral makes sense.
3.1.2. Complications Arising in Practice
Up to now everything looks nice, and an especially neat aspect is
that we obtained Abel's equation by a very easy geometrical argument.
Physical reality, however, presents additional difficulties, some of
which will be described below.
At first,the model should be critically evaluated. There may be
internal absorption within the cylinder, and this absorption may even
depend on the intensity g(r). One then arrives at a nonlinear integral
equation. For the-sake of simplicity,let us adhere to the linear model.
There are no serious problems if the data function g is available
with, high accuracy at sufficiently irany equidistant points. There is a
large arsenal of effective discretization methods from which to choose
an appropriate scheme.
But often G is contaminated by noise ( inaccurate measurements or even
noise in the physical experiment itself, e.g. perturbation of symmetry)
so that (at best) we have not G itself but a function
C.1.5) G(x) = G(x) + p(x), 0<x<R .

38
On the structure of the perturbation p , the noise, various assumptions
can be made. In any case, noise is amplified by using one of the
inversion formulas C.1.2) or C.1.3) or a discrete analog , because Abel's
integral equation is ill-posed in customary function spaces: solving
it corresponds to differentiation of order a, and here we have a = 1/2.
We do not recommend here the classical Tikhonov regularization
(see, [Ti-Ar,1977]) which is based on the idea that the true solution
is rather smooth and should therefore be estimated by minimizing a
suitable quadratic functional containing the discrepancy and a
regularization term. We prefer to exploit extra qualitative information
ong , expressed via inequalities (compare [Go-Ko,1966], [La,1969],
[Go,1979]). That is, we look for a function g, hopefully near g,
whose transform G via C.1.2), in a sense to be specified ,best
fits the (noisy) data ,thereby respecting the extra information
available.
Often one or more of the following types of extra information are
at hand (for 0 < r < R) .
(a) g(r) > 0. (b) 0 < g(r) < c .
A
(c) g (r) increasing (g' (r) > 0 or even 0 < g' (r) < c) .
(d) g(r) convex.
(e) g(r) unimodal, at first increasing, then decreasing, with an unknown
argument r such that g(r )= max g(r).
max max n „
0<r<R
g may even have a sharp peak whose location r is of interest.
R
(f) 2n J g(r) r dr is given (total luminosity).
0
A quite natural assumption is g'@) = O.In physical space
I 2 2 A
g(r) = g(yx +y ), and the graph of g(x,y) = g(r) is smooth at x = y = r = 0
only if g'@) = 0 .
As a final remark, we say that the general theory (the pure analysis
as well as the approximation schemes) should include the case of the
solution g being a special type of generalized function, namely a measure.
If the luminosity is concentrated in a narrow vicinity of one or a few
values of r, then g may appropriately be modelled by a linear combination
of 5-functions. See [Go,1937].

39
3.2. Stereology of Spherical Particles
3.2.1. The problem
The general problem is to determine the probability density of the
sizes of particles in a solid opaque medium from that of the sizes of
their intersections with a random plane cut. Applications are, among
others, the study of corpuscles embedded in the (solidified) tissue
of a biological organ ([Wi,1925], [wi, I926]),in sedimentary petrography
[Kr,1935], in crystallography [Sc,1931], in investigation of carbide
particles in steel ([An,1930], [An-Ja,1974], [An-Ja,1975 and 1975 ]) ,
concerning metallography see also [Fu,1953] . we especially recommend
the lucidly written paper [Re,1955] .
For simplicity and in order to arrive at the classical Abel
equations, we assume the particles to be A pke.1-leal. However, ellipsoidal
particles have already been considered by [Wi,1926], cylindrical ones by
[Fu,1953] . For the general theory of convex particles, we
refer the reader to [Sa,1955], [Sa,1976]. See also [ Bad,1932]for a
treatment of more general stereological problems.
In the sequel,we draw from the highly readable original paper of
[Wi,1925] and on the description given in the book by Kendall and
[ Ke-Mo, 19.6 3] , but with a change of notations (taking the independent
variables r a.nd x to be radii of spheres and circles, respectively,
instead of diameter, and denoting probability density functions by
small letters) which does not change the form of the integral equation.
Assume the following model: Spherical particles (called "spheres")
3
are randomly distributed in IR , more precisely: their centers are
assumed to be distributed according to a Poisson field with unknown spatial
density
C.2.1) A = mean number of centers in a volume of size 1 .
The radii r of these spheres obey an unknown probability density f(r)
where 0 < r < R < °° or 0 < r < R = °° and R is an upper bound on the
possible size of a radius. Note that we do not allow spheres with
radius zero. We should have
R
f(r) > 0 and J f(r)dr = 1 .
0
We want to itieii, however, that regardless of our way of speaking of
f as a density,we may consider f as a ge.ne.ial-ize.d fiunct-Lon in the sense
of Gelfand and Shilov A964) . If, e.g., f is a linear combination of so-
called 5-functions we have the discrete probability case for r, meaning

40
that r can assume only a discrete set of possible values (with
probability 1). In this more general case the reader may replace f(r)dr
by dF(r) in the following formulas if he does not want to abuse the
integral sign.
The model described amounts to neglecting the possibility of spheres
overlapping each other and is, roughly speaking justified if the mean
distance between centers is much larger than the mean radius
R
C.2.2) rQ = J r f(r)dr
which we always assume positive but finite.
Fig. 3.2.1 View along cutting plane
Let now E be a random plane. With probability 1 this plane does
intersect some of the spheres. Denote by r the radius of a hit sphere
and by x the radius of its circle of intersection with E (see Fig. 3.2.1) .
We call r the "actual radius", x the "apparent radius". The random
variable x will be distributed according to a probability density g(x),
0 < x < R < °o or 0 < x < R = °°, with g(x) >0 and j g(x)dx = 1 .
0
There should be no misunderstanding if in the sequel we always
write < R, even in the case R = °o.

41
By counting,a statistician can estimate the density g, and
assuming he has done this job well, we find ourselves faced with the
problem of calculating the unknown probability density f of the actual
radius r from the known one, g , of the apparent radius x. We often
need a few mean values, namely the average spatial density A
(see C.2.1)) of spheres, their mean actual radius r = M.. (see C.2.2))
their mean surface 4 tt M~ , and their mean
volume -=- M.,
C.2.3)
Here we use the notation
M. = J r3f(r)dr,
D 0
= J x^g(x)dx
for the moments of the densities f and g, the index j running through
the integers 0,1,2,3,..., in addition to the value -1 for m.
(this special case will be needed later). we shall see that in those
instances where one is interested only in these global quantities the
calculation of f may be circumvented by direct determination of the
moments M. from the moments m. which fortunately can be done in a very easy
way. The meaning of the moments M? and M, should be clear: M~ may be
proportional to the chemical or physical activity of the spheres, M3 is
proportional to their mean mass if they all consist of the same homogeneous
material.
3 . 2 . 2 . Formal Solutions
To solve the problem, we are now going to show that for f assumed as
given, g can be obtained by an Abel-type integral transform, i.e. we will
establish an Abel-type integral equation for the determinaticnof an unknown
density f from a known one, g . We will proceed by differential arguments.
unit area A of E
Pig. 3.2.2 Sketch for geometric probability

42
The expected number of spheres with (actual) radius in (r,r+dr]
which intersect E so that the center of the cutting circle lies within
an area A of size 1 is (see Fig. 3.2.2)
C.2.4) A • 2r - f(r)dr.
Thus the probability density f* (r) for a sphere intersecting E having
(actual) radius r is proportional to r f(r), hence for scaling reasons
R
(J f*(r)dr = 1 should be fulfilled)
0
C.2.5) f*(r) - r £(r)
r
o
Now take a sphere with radius r and let y£ [0,r].
Then under the condition that this sphere is cut by E,the probability
density of its center having a distance y from E is piecewise constant,
namely
C.2.6) i for y£ [0,r], 0 else.
/~2 2
By Pythagoras (with x as apparent radius) y = vr -x for 0 < x < r, so,
given r, y is decreasing in x and - -s~- - x
ax /~2 2
Vr -x
Hence (see 3.2.6) the event that a sphere of actual radius r,
cutting E, has an apparent radius x, obeys the probability density
C.2.7) -!<&=! *_ , o<x<r .
r dx
r P~
vr -x
2
Multiplying by f*(r) and integrating over x<r<R (those values of r
contributing to a value x of the apparent radius) we obtain
/-,-,o\ x r f(r)dr ,. _ _
C.2.8) — J —^—3 = g(x), 0<x<R .
o r=x /~2 2
Vr -x
We check that g actually is a probability density.
Trivially, g(x) > 0 for 0<x<R. Furthermore

43
R
J g(x)dx =
0
- J rX / f(r)dr
x=0 o r=x / 2 2
yr -x
1 R
= —■ J f(r)r dr = 1
° 0
dx = -L J f(r) J X-^L_ dr
o r=0 x=0 /~2 2
Vr -x
The inverse problem now is the following. Given the probability
R
density g(x), 0<x<R, where g(x) >0 and J g(x)dx = 1 , determine the
o
probability density f(r), 0 < r < R, and the mean radius r (see C.2.2))
from the integral equation C.2.8) and the scaling condition
R
C.2.9) J f(r)dr = 1 .
o
To solve this problem we observe that C.2.8) is an Abel integral
equation of type C.a) for the determination of an unknown function
f(r) = f(r)/r from a known function g(x)/x. By formula C.b) we get
C.2.10) lU) = - f £ J ^^ , 0<r<R ,
n ar r /2 „2
-r
and then set
,R -1
C.2.11) ro=f/?(r)drj , f(r) =rQ¥(r),
to fulfil the scaling condition C.2.9) .
We shall postpone checking fulfilment of C.2.2) to the next
section 3.2.3 .
The mean spatial density of (centers of) spheres can now be obtained
by integrating C.2.4) over 0 <r <R. This yields A•2r as the mean number
in a unit area A of E of circle- centers arising from intersecting spheres
and can be estimated by a statistical counting method, r being given by
C.2.11),we get an estimate for A .
3.2.3. Is the Formal Solution Correct ?
We first remark that from g being a probability density it does not
follow that r'(r) >0 everywhere. An easy calculation shows, e.g., that if
g(x) = 5(x-1) in case R> 1, 8 being Dirac's delta-function, then f(r)
is not everywhere > 0. Thus our solution procedure may produce a
function f which is not a probability density, and in actual practice

44
this may happen if the data function g is contaminated by noise or is not
estimated with sufficient accuracy.
We should take into account the smoothing properties of the
transformation C.2.8), applied to a probability density f. This smoothing
has the property of half order integration and thus, certainly, not
every probability density qualifies as g. One necessary
condition g must satisfy is g(x) >0 in a positive vicinity of 0. There is
R
a positive number p such that J f(r)dr>0, and C.2.8) now implies for
0 < x < p the inequality p
g(x) >
& 2
VR -p
J f(r)dr > 0
P
Let us now check that C.2.2) is satisfied ; but this follows from
C.2.11) if
C.2.12)
R
J r i"(r) dr = 1
0
By C.2.10) we have
R
r i i \ j 2 r d r g (x) dx ,
J r f(r)dr=-_ J r -=— J -^-5—- dr
11 r-0 dr x=r v£7-2
r j
g (x) dx
/~2 2
Vx -r
R
r=0 r
J J g(x)dx drl
=0 x=r /~2 2 J
Vx -r
and if the conditions
C.2.13)
lim r J SJjOdx = Q ^
r-+R x=r Al
2 2
-r
C.2.14)
g(x)
dx < °o
are satisfied we can conclude that
J r f (r)dr = £ J g(x) J
dr
0 v^-r2
dx
9
- / g(x) 5 dx = 1
So, at least under the extra conditions C.2.13) and C.2.14) we
have C.2.2) (even if the calculated function f is not everywhere
positive, i.e. is not a probability density).

45
What is the relevance of these extra conditions ? We shall show
that C.2.14) is necessary for g to stem from a probability density with
0<r < <x>; and that C.2.13) is necessary under the additional condition
R < °o .
Assume g related to a probability density f according to C.2.E
From
we find
tt/2 = J (r -x
0
2 2,-1/2
dx
jMl f(r)dr = ; ;
o r=0 x=0
dx
n.—j
Vr -x
f (r)dr
R R , , . , R r g(x)dx
= ; ; f_(r)dr dx = ; o^ ^
x=0 r=x Jp—2 0
R . .
the last " = " being implied by C.2.8) . Thus ^ = r0 / 2-^- dx ,
0
R
C.2.15)
J
g(x)
dx =
2r
So we see that C.2.14) is valid.
Furthermore C.2.10) and C.2.11) imply
1 = r J ?(r)dr - - 1 r J 2i2Li^
O J y TT o J .
2 2
v£
R
r=0
» 2 r { J SLi*i dx _ lim j 2iil_^ I
n ° I 0 x r-+R r A2~T J
£
and using C.2.15) we conclude that
R
C.2.16)
l ;m r g(x)dx _ .,
lim J '■ = 0 ,
r->R r / 2
Vx -r
hence that C.2.13) must be satisfied under the additional assumption
R < ».
In the case R = ^ the condition C.2.13) is fulfilled if g(x) -+0
— R
sufficiently fast as x-+°°. Assume, e.g., that g(x) < Ax with A > 0,
8 > 1, for x > x*. For r > x* we then have

46
oo oo oo — K
r j g(x)dx = j r g (rt)dt < Ar+1 j t dt
r ^7 1 v^T" ~ 1 ^7
In applications two kinds of -Lll-po6e.d-ne.66 are to be considered.
(a) The sensitivity of f to inaccurate data g.
(b) The measured function g, which should lie near g, may not lie in
the range of the transformation C.2.8) applied to admissible f.
Not only must g be so that everywhere f(r) >0 but also conditions like
C.2.14) and C.2.13) or even C.2.16), the latter in the case of finite
R, should be met. In the development of numerical methods, these
conditions should be taken into account for regularization if the data are
seriously contaminated by noise. See e.g. [Vi-Go,1933] for an algorithm
taking account of the nonnegativity of f.
There may be additional shape information on f available which can
be used for regularizing purposes.
3.2.4. Calculation of Moments
Statisticians like to describe a probability distribution by its
moments, defined in C.2.3)
R
C.2.17) M = J rJf(r)dr, j = 0,1,2,...,
0
R
C.2.18) m = J xDg(x)dx, j = -1,0,1,2
3 0
As already said ,the moments M.,M?,M, have simple geometric meanings, the
most important in our formal considerations being
C.2.19) M1=r = mean actual radius .
It will be very convenient also to have available the moment m_... In
section 3.2.3 we have seen that m is finite for a density g arising
from an admitted probability density f, and C.2.15) can now be
translated into
C.2.20) 2m_1 M1 = 2m_1 rQ = tt .
For n> 2 we have, using C.2.8),

47
nn , - f J xn J fi£^£ dx = f J J 5^^- f (r)dr
° 0 x v/r2-x2 ° 0 0 /r2_x2
Substituting t = x/r and then t = cos u we get
r n^ „1 ^,, tt/2
r x dx n r t dt n , . . n , n
J ——^^2 = r J —zm: = r J (cos u) du =: I r ,
0 \/r2-x2 0 /^
2
0
n
hence, taking account of C.2.20) ,
I R 2m ,
m . = — f rnf (r)dr = I m
n-1 r ' rt n n
o o
C.2.21) M = -^- I„1 ni . for n>2
n ^m_i n n-1
As is well known,
C.2.22)
*.-{
n
2
1 ■
1
2
2
3
3
4
4
' 5 ' •
n-1
n
n-1
n
if n is even
if n is odd.
Together with C.2.20) which can be transcribed into
C.2.23) Ml = ^
the moments M. of the unknown density f can,by C.2.21) and C.2.23) ,be
very-easily calculated from those of the experimentally determined density
g. Note, however, that the M's are very sensitive to the variations of
R , \
m = J <Li*l dx .
-1 J x
0
So special care has to be taken in determining the density g near zero.
We remark that C.2.23) is a special case of C.2.21), namely for
n = 1 (I1 = 1, mQ = 1) .
More generally (compare [An,1978]) consider a linear functional ip,
defined on the admissible densities f, and having a bounded and differen-
tiable representer which we, for convenience, denote by the same letter
ip»
R
C.2.24) <tp,f> = / ip(r)f(r)dr .
0

48
Then in view of B.2.10), B.2.11), an integration by parts and a change
of the order of integration give
C.2.25) <ip,f> = —2 |(p(o)m , + // ^ (r)dr g(x)dx[
n v ' x=0 r = 0 /~2 2 J-
yx -r
This formula allows us to compute linear functionals acting on f by computing
corresponding linear functionals acting on g (at least if ip is bounded and dif-
ferentiable).
3.2.5. Final Remarks
There is a special class of probability densities that are the same
for the actual and for the apparent radius.
For 2
f(r) = —=• exp(- -^-p) with a > 0, R = °°,
a 2a
2 2
we find r = o\/tt/2 ; and in view of B.2.3), using the substitution 2t=r -x
and the definition of the gamma function , we get
2
g(x) = -% exp (- ^) = f (x) .
a 2a
Interesting aspects are also discussed by [Reid,1955].
He arrives immediately at Abel's classical equation by considering appar-
2 2 .
ent areas a = nx and actual central cross-sectional areas A = nr instead
of apparent radii and actual radii. He furthermore shows that the density
of the apparent radii is given by an Abel equation involving the density
of the length of half-chords cut out by a random line. He finds that by
combination one can determine the density of the actual radii from that
of half-chords by applying twice in succession a half-order
differentiation process; hence by an ordinary differentiation process.
A nice generalization of spherical stereology is treated in papers
of [Ba,1959], [Ba,1965], [Go,1967] , namely the ■'tomato salad problem".

49
Pig. 3.2.3 Tomato salad
A random slice bounded by parallel planes with thickness 2s > 0
is cut out from the solid opaque medium . The parts of spheres cut
out by the slice are projected orthogonally on another plane (parallel
to the slice) where they give rise to observable radii. The slice is
assumed so thin that overlapping of "shadows" can be neglected. If now
the probability density of the oberserved radius x is denoted by g(x),
then the probability density of the actual radius r is uniquely given
via an Abel integral equation of the second kind:
C.2.2(
sf (x)
(s+rQ)g(x) - x /
f(r)dr
0 < x < R
x ,L2 2
vr -x
Goldsmith A967)gives an explicit solution formula for this equation.
The "tomato salad problem" is well posed, because it leads to an
Abel integral equation of the second kind. Note, that for s-+0 equation
C.2.26) tends to equation C.2.8).
As an exercise,we propose to the reader to consider the case of a
discrete probability distribution of the actual radius in the problem

50
treated in 3.2.1 to 3.2.4: Let the r, e @,R) be distinct for j £ S,where
S is a countable (= finite or countably infinite) set of indices. Let
all a. > 0 and
J -
I a. = 1, f(r) = I a. 6(r-r.). Then r = Z a. r.
jes => jes 3 3 ° jes 3 :
Show that formula C.2.8) yields
a .
g(x) = f- T J
o j I r.>x /2 2
1 vr.-x
and that the g(r.) can be so defined that g is left-continuous.
R
Furthermore show that J g(x)dx = 1 and that f can be recovered from g
0
by the method described in 3.2.2
Finally the reader is advised to imagine the case where there are
also spheres of radius 0 (mass-points embedded in the opaque solid medium).
Looking at C.2.4),he should convince himself that only with probability
0 are they hit by a random plane. Indeed:
J A • 2r 6(r)dr = 0 for any e > 0.
[0,e)
This amounts to the fact that they cannot be detected, and we have been
wise to ignore them. In practice, however, it means that f(r) cannot be
determined very accurately from experimentally estimated g(x) if r is
very small.
The tomato salad problem, however, can be modified so as to
treat effectively this excluded case of zero radius.
3.3. Inversion of Seismic Travel Times
3.3.1. General Considerations
In the first decennium of this century ,Benndorf A906), EerglotzA9Q7)
and Wiechert [Wi-Zoe,1907] developed a model for investigating the
earth's interior by measuring the time seismic waves, generated by an
earthquake or an artificial explosion, need to travel underground from
one surface point to others. One wants to obtain
information on the elastic characteristics of the sub-surface medium, and
these being related to the local velocities v and v of pressure waves
(these are longitudinal) and shear waves (these are transversal),respec-

51
tively, one is interested in a method of determining these
characteristics by suitable measurements on the terrestrial surface. According e.g.
to [Bu,1963] we have
v = \/( X + 2y) /p , vs = vWp
where p denotes density, and X and y are the Lame parameters.
Considering one type of velocity, calling it v, Benndorf, Herglotz
and Wiechert assumed the laws of geometric optics (the Snellius
refraction law) to hold and the earth to be radially symmetric (i.e. all
properties to depend solely on the distance r from the earth's center).
They further assumed that v = v(r) strictly increases as r decreases.
By suitable manipulations and transformations of variables,they obtained
an Abel integral equation of type C.a) whose solution allows toi£ to
compute v(r). For highly readable accounts of the problem and its
treatment see, e.g., the books [Bu,1963] and [Ga,1971].
The model often is realistic if one does not want to get
information on very great depths and if the region investigated is not too
large. More complicated models must be used if radial symmetry cannot
be assumed or if there are "low velocity zones" below the surface, zones
where v(r) (as r decreases) is decreasing instead of increasing. The
general problem may be called the problem of "tomography of the earth's
interior".
In recent decades much has been published on these problems,
especially on their computational treatment. The literature on the subject
is impressive; we refer the reader to the papers of Garmany,
Orcutt, Parker, Mackenzie and McClain, written between 1976 and 1980,
Keilis-Borok A972) ,Johnson and Gilbert A972) ,Bessanova, Fishman,
Ryaboyi and Sitnikova A974) and S. Campi A98C),to name but a few.
Here we shall not describe the spherical earth model but the "flat
earth model" thereby avoiding some purely geometric problems hiding the
essential ideas. The flat earth model is appropriate for studying the
sub-surface structure within not too great depths and below a not too
large surface. This flat earth model is treated in several of the above
mentioned papers, and [Ke-Bo,1972] shows how by a transformation of
coordinates and functions the "round earth model" can be reduced to it.
3.3.2, The Flat Earth Model
We assume the surface of the earth as a plane and denote by z the
magnitude of depth below this surface. We further assume that the sea-

52
lar velocity v of a seismic ray does only depend on z, and that this
velocity v(z) is strictly increasing and continuously differentiable
for 0 < z < z , where z is the maximal depth to be investigated.
- max max ^ ^
Since we just want to give an introduction (the reader interested in
all the complications that can arise in practical situations should study
geophysical literature) we treat only the case where
C.3.1)
v' ( z) > 0 for 0 < z < z
A seismic ray, emanating from adefinite surface point (x=0 in
Fig. 3.3.1)penetrates along a curve (by refraction according to
geometric optics) into the subsurface medium, and if it is sufficiently
strongly bent it reaches a
x=0
x=X/2
x=X
> x
Pig. 3.3.1 A ray trajectory
lowest point z = z from which on it turns upwards again reaching the
surface at a distance x=X from its source. Knowing the moment of its
emanation from x=0 (this moment is precisely known in the case of an
artificial explosion) the moments at which rays come up again at various
surface points can be measured,thus yielding a travel time T = T(X).
A
Knowing this function T(X) for 0<X <X, one wants to determine v=v(z)
A A
for 0 < z < z, where, of course, also the value of z is not known a priori.
To analyze this problem, we first take as granted the existence of a
velocity v(z) for 0 < z < z with the properties described and then de-
2 -- max * *
rive, a relation between the functions v(z) and T(X), in fact an Abel
integral equation. We must investigate the properties of the trajectories of
rays.

53
Consider a definite ray and its trajectory
x = x(t) , z = z(t), 0<t<T, x@) = z@) = 0, where t measures time,
T being the finite travel time if the ray reaches the surface again
(otherwise T = °° and 0 < t < T) . At any point (x,z) of the trajectory let
i = i(x,z) be its angle of incidence with the vertical line through
this point (tacitly assuming the trajectory to have a tangent). The
basis of our consideration now is the Snellius law of refraction. Along
the trajectory of a ray (sin i(x,z))/v(z) is constant:
C.3.2)
sin i(x(t) , z(t)
v(z(t))
where p, the "ray parameter", depends only on the ray chosen. We note
that in particular
C.3.3)
sin i @,0) _ sln 1o
v@)
i = i@,0) being the ray's initial incidence angle (on the surface) and
v = v@) the velocity on the surface.
From C.3.1) and C.3.2) we can draw important conclusions.
>x
Pig. 3.3.2 Configuration at surface
We have 0 < p < 1/v , in particular p = 0 for the vertical ray (this ray is
not bent, its trajectory is the straight line perpendicular to the
surface),p = 1/v for the horizontal ray (likewise a straight line).
-]
For a ray with parameter p in the open interval @, —) C.3.2)
vo
tells us that as long as the ray travels deeper (z increasing) sin i=pv
increases from sin i on upwards, and hence also i increases from i on
0 r • o
upwards. If after a finite time t has elapsed, the ray reaches a point
P

54
(x , z ) with i(x , z ) = Ti/2,then it turns upwards again, and for
reasons of symmetry it reaches the surface at X = 2x at the moment T = 2t ,
p p
and T is its travel time.
The second half of the trajectory is obtained as mirror image of the
first half with respect to the vertical line x = x . In any case, the
r p
trajectory of a ray can be written as a function z = z(x), and this
function is concave by the aforementioned property of increasing
incidence angle i.
We observe, looking again at C.3.2), that our ray with parameter
pE @,1/v ) cannot have a horizontal tangent (sin i = 1) if v <v(z) <1/p,
hence after leaving its origin at x=0, z=0 it must penetrate deeper and
deeper (z must increase) as long as v(z) < 1/p. If there is a value z
with v(z ) = 1/p, i.e. if v(z) reaches this value, and if there exist
finite values t and x such that x(t ) = x , z(t ) = z , then the tra-
p p P P P P
jectory reaches a deepest point, the tangent in this point is horizontal,
the ray turns upwards again, and we have finite values X = 2x and
T = 2t , belonging to our value p.
We shall now show that such finite values t and x indeed exist
P P
(provided 1/p is a value taken on by the velocity v for a value z=z ).
For the trajectory x = x(t), z = z(t) of the ray with parameter p
we have the system of differential equations
dx d z
C.3.4) ^- = v(z)sin i(x,z), -^ = v(z)cos i(x,z)
with initial condition x@) = z@) = 0 and, by C.3.2),sin i(x,z)=pv(z) .
It follows that, as long as 0 < z < z ,
dx _ pv(z) dt _ 1
dz ~ , j ' dz ~ I 2
/l-(pv(z)r v(z)/l-(pv(z))z
We have to show the finiteness of both values
z z
fP pv(z)dz . _ ? dz
C.3.5) x = J *"**'"* , t = f
p J p J
o vV(PV(z)r u v(z)v/i-(pv(z)r
To this purpose we use the assumption C.3.1). In particular we have
v1(z )> 0 and v(z ) = 1/p, and we find

55
1-(pv(z)J= A-pv(z))A+pv(z)) = (pv'(z )(z-z )+a(z-z ))A+pv(z)).
So both improper integrals are convergent, x and t are finite.
For further treatment,it is convenient to substitute
C.3.6) u = 1/v .
Note that u = u(z) decreases from u = 1/v downwards, remaining
positive, and that u'(z) =- v (z)—^ < 0 for all z>0 .
(v(z)r
With this new variable we can state as result
Theorem 3.3.1 I fi tkeie -ii, a value, z {,oi wklck u(z ) = p (wkeie 0<p<u )
then tke lay w-itk paiametei p leacku tke itDi^ace aga-in at x = X w-itk fa-in-ite
tnavel time T, and
z z ?
C.3.7) X = 2 /P P dz , T = 2 f (U(Z)) dZ ■ .
0 v/(u(z)J-p2 ° \/(u(z)J-p2
We thus have X = X(p), T = T(p) as functions of p, however, these
functions need not be monotonic. See, e.g., the discussion in [Ga,1971],
It may happen that as p begins to decrease from 1/v = u downwards,
X or T first increasesthen begins to decrease for rays penetrating deeper.
However, we can introduce the so-called delay-time function
C.3.8) T(p) = T(p) - p X(p)
z
2 J /(u(z)J-p2 dz
which is monotonic. Indeed, assuming sufficient smoothness,
z
p P
i'(p) =-2/ V - dz = - X(p).
0 /(uf—2 -2
i(z)) -p'
From this and from C.3.7) we find, formally,
,., dT dx „. .
T (P> = dp" " P dp" " X(P>
dT dX ,.,
d? " p dp" + T (p> '
hence locally

56
C.3.9) p = g,
so that p can be determined as the slope of the travel-time curve.
Introducing now the function z = z(u) inverse to u = u(z) and
using the integral representation in C.3.7) we get
—4—- du and, integrating by parts,
T(p)
T(p)
= 2 ;
u
o
u
o
= 2 ;
p
/ 2 2
vu -p
u z (u)
/ 2 2
vu -p
C.3.10) T(p) = 2 J u ^vu' du .
P P- 2
VU -p
The inverse problem is the following. Given sufficiently many pairs
of measurements (X,T), we have to determine for each pair its parameter p
(this cannot be measured directly) from C.3.8). This yields the
functions X(p), T(p), hence by C.3.7) r (p) .
If we now have x (p) for u > p >p*, we can solve the Abel integral
equation
u
r u z(u) , 1 ,,
J -—= du = ? x(p)
P vC2 2
P
for z = z(u), where u >u>p*, and obtain u(z) as function inverse to
z = z (u) .
Noting now that at the deepest point z = z of a ray we have i=n/2,
hence by C.3. 2) u = — = p, we see, using the monotonicity of the
functions u = u(z) and z = z(u), that we can obtain u(z) and hence
v(z) = 1/u(z) for 0< z< z* where
C.3.11) u (z*) = p* .
3.4. Refractive Index of Optical Fibres
In recent years,optical fibres (or glass fibres) have become more
and more important as a means of transmission of signals, and this trend
will certainly continue. In the manufacture of such fibres
non-destructive methods are needed for measuring their optical quantities, i.e.
the refractive index n as function of the radius r (the distance from
the central axis, the fibre being assumed as a long circular cylinder
produced by deposition of successive layers of doped silica within a
rotating silica tube).

57
We describe in some detail a method to determine the refractive
index under the assumption that this index varies only il-igktly
within the fibre (compare Marcuse A979)). A more complicated method (under
less restrictive assumptions) gives more information and is treated and
discussed by Shibata et alii A979), see also Anderssen and Calligaro
A981). This "Japanese method" uses the photoelastic effect by
illuminating by laser light the fibre which by elastic stress is optically
anisotropic. The phase difference (retardation) of the two orthogonal
splits of the laser ray is measured, and again one arrives at Abel type
equations for the refractive index. The restrictions are much less severe
than in Marcuse's method, the mathematics is much more tricky; however
even discontinuous refraction indices can be determined (for example step
functions, which are often relevant in practical situations).
However, we refer the reader to the quoted papers for this intricate
method,and are now going to describe Marcuse's simple model.
He puts the fibre into a homogeneous liquid whose refraction index
n matches that of the fibre surface so that there is no jump of the
index n at the fibre boundary.
He assumes that within the fibre,the refraction index n(r) varies
only slightly (depending on r) and as already said n = n(a). Thus a
ray is bent only by a small angle, after leaving the fibre
incident parallel
klight with constant
intensity
Fig. 3.4.1
>x
E= plane of observation

58
the ray is again straight. There is a plane of observation orthogonal
to the direction of emission of light rays on which for each ray emanat-
A
ing at height t,its height y(t) of incidence can be determined from the
fact that on the plane of observation the total power received between
A
y = 0 and y = y(t) is equal to -that between y = 0 and y = t to the left
of the fibre (see Fig. 3.4.1 ). The latter is a known linear function
of t.
Of course, it is assumed that the plane E of observation is not
too far off from the fibre. E should be so near the fibre that different
rays do not cross each other before reaching E and that different rays
hit E at different points. However, L should be so large that a << L,
where a is the radius of the fibre, L the distance of E from the center
of the fibre.
Since n(r), 0<r<a, varies only slightly, any ray trajectory
y = y(x) within the fibre can be described by the paraxial ray equation
(consult text-books on geometrical optics) in good approximation
C.4.1) 4=1 r
dx2 nc ^
Outside the fibre,the ray is straight with slope 0 to the left of the
fibre, with a slope equal to that of its exit point to the right of the
fibre. A typical ray trajectory, namely one starting at a height t with
0 < t < a (thus meeting the fibre; for symmetry reason we can restrict our
attention to nonnegative values of t) consists of three parts 1.,1-,1-.
T1 : y(x) = t for x < - Va2-t2 = x*(t)
1j : y(x) obeying C.4.1) in x*(t) < x < x(t)
with entry conditions y(x*(t)) = t, y'(x*(t))= 0
and x(t) as abscissa of exit point .
T : y = y(x(t)) + y'(x(t))(x-x(t)) for x(t) < x < L
3
We can calculate the exit slope of the ray by help of C.4.1) as
y«(x(t„ .lj gdx =1 J n'(r) |f dx
c T_ 1 c T_ J
J_ J n.(r)Z.£_dr = ! ; y_
n L r x n ' y x
c T2 c T2

59
= ^ J n' (r) y dr .
nc t0 n.—2
2 yr -y
Denoting by r the closest distance of T^ to the origin, the inte-
° 2 /2 2~~
gral can be split into two parts (with r = r(x,y(x)) = vx +(y(x)) )) .
r
o a
J =-/ n' (r) y dr + J n' (r) y dr ,
T- a /~2 2 r /~2 2
2 vr -y o vr -y
the first part, with dr< 0, coming from the ray before reaching its
closest point to the origin, the second part, with dr>0, coming from
the ray after having traversed its closest point. Hence, by symmetry,
9 a
C.4.2) y'(x(t)) = -f- J n'(r) y dr .
c r /~2 2
o \Jr -y
At this place flarcuse drastically simplifies things by exploiting
the stated smallness assumptions. With n(r),and hence y(x) , varying only
slightly within the fibre,he feels justified to replace C.4.2) by
C.4.3) y' (x(t)) = —■ \ n' (r4 dr
using y«t.
A
The height y(t) of the point of incidence of the ray on the plane
E of observation is
y(t) = y(x(t)) + (L-x(t)ly1(x(t)) .
Using a<<L,and accordingly x(t) <<L, and C.4.3), Marcuse replaees this
by
y(t) = t + L £ ; ^i£l^ ,
which is equivalent to the Abel equation
a n'(r)dr = y(t)-t n
t v^2^2 2 L t C
The solution formula A.B.6i) gives
«'^ -£ -hi -7=^ '

60
hence we have, considering n(a) = n ,
n(r) = n U + ± ) ^-^ jdt .
This formula gives us the refraction index n(r) as an Abel type
transformation of the measured function y(t).
It is important to note here that this problem is wnll-po&nd.
Determining n'(r) is -M-po6e.d, corresponding to half-order
differentiation by solving Abel's equation, but. to 9et n(r)i a further
integration is to be carried out. So, to determine n(r) from given y(t) corres-
3
ponds to an integration of order -^ and thus has a good smoothing effect on
possible noise inherent in the values y.

61
Appendix 3As Linear Generalized
Abel Integral Equations
We here give a glimpse into the topic of geneia£-t'zed Abe.1 e.qua-
t-ioni. The two main types are
x b
C.A.1) Mu(x) := fj|j- j (x-t)"-1 u(t)dt + fj^j- / (t-x)" u(t)dt = f (x) ,
a x
x b
C.A.1i) M*u(x):= jr^j- / (x-t)™" (u<j>) (t) +r^y / (t-x) a_1 (u<j>) (t) dt = f (x).
Here a < x < b, the functions ¢,^, f are known, 0 < a < 1, and u is the
unknown function. We suppose that | <)>(x) I + I,ip(x) j * 0 for a < x < b .
Generalized Abel equations or systens of such equations arise in neny
problems of mathematical physics, elasticity theory, see [Lo-Wa,1979] ,
[Li,1967], [Wa,l965], hydrodynamics, [Ch-Lu,l959] and partial
differential equations, see [Wo, 1965] .For a numerical treartment, see [Li-No, 1 971 ] .
In elasticity theory, for example, one considers the problem of
determining the stresses in an elastic half-space L under the influence
of a normal pressure applied by a rigid punch of circular cross-section.
Knowing the contact radius a, the total penetration 6, the total applied
load (see Fig. 3.A.1), the deformation u (r,0) of L in the direction z
and the radial deformation u (r,0), we want to find the normal stress
p(r) and the tangential stress q(r).

62
^ j-»
Fig. 3.A.1
We obtain the following system of generalized Abel equations:
C.A.2) C. J r(r2-x2)_1/2p(r)dr-C9[/ q(r)dr-x J (x2-r2)~1/2q(r)dr]=u (x),
x 0 0
C.A.2i)
J (x'
•1/2
2 2,-1/2
rpfrJdr+CjX / (r -x ) ' q(r)dr = u2(x) ,
where C. and C? are two constants depending on a and 5, u1 and u~ are
two known functions, determined oy uz and ur . The methods used for solving this
system ([Gi-We 1976], [Lo-Wa,1979],[Li ,1967], [Wa,1979]) are similar to
those used for solving the equations C.A.1) and C.A.1i). We shall
discuss them in Appendix 6.A .
Generalized Abel equations arise also in the study of mixed problems
for Tricomi equations, see [Wo,1965], We consider, as an example, the
half-space boundary value problem to find solutions u of
C.A.3)
y u (x,y) + u (x,y)
2 xx J yy
o,
<x<+ °°, y>o,
C.A.3i)
C.A.3ii)
C.A.3iii)
u(x,0) = t(x)
u (x,0) = 0
u(x,y)
0 < x < 1 ,
x < 0, x > 1 ,
I (x,y) I -» «°

63
By using the Laplace transform we can prove, see [Ch-Lu,1959], that a
solution of the equation C.A.3) with the condition C.A.3iii),
considering u (x,0) as data, is given by
+°° -2fi
C.A.4) u(x,y) = C6 J Iz-CI u (CO)dC ,
where z = x + iY, X =^ y U+2) /2 ,
_ d-*gJB rn+6)
2(m+2)' ^6 "
26/t r(I+6>
Now, frcmC.A. 4) and C.A.3i),it is clear that for solving the complete
problem we should find V(x) = u (x,0) in [0,1]. In fact, by C.A.3i)
u (x,0) is known in (-00,0) U A,+°°) , and if we take account of u (x,0)
in [0,1] we can use formula C.A.4) for finding u. By C.A.4), C.A.3i),
C.A.3ii) we obtain for v the integral equation
1
6
C.A.5) t(x) = CR / |x-Cl 26 v(C)dC
0
This is an equation of type C.A.1) with <j> 3 i(j = C„ -
Carleman was the first to study equation C.A.5),
see [Ca,1922]. He did this by transforming it to a Hilbert-Riemann
boundary value problem [Ga,1966]. His method was used by Sakaljuk, in 1960,
for solving the equation C.A.1) with if and ty non-constant. We briefly
discuss this technique and other solution methods in the Appendix A to
Chapter 6.
Remark: Whereas in all previous applications the exponent of the
jlarity has been a = 1/2,we new have exponent 7
ral is different from 1/2 (equal to 1/2 only if m
singularity has been a = 1/2,we new have exponent 28 = ——y which in gene-

Chapter 4 ; Smoothing Properties of
the Abel Operators
This chapter is dedicated to the description of trie more important
smoothing properties of the Abel operator
(Jau)(x) = YTZ) / (x-t)a~1u(t)dt, 0 < x < a .
o
The case 0 <a < + «> is treated in detail. However, many properties
hold also if a = t » .
The basic smoothing properties, presented in paragraphs 1 and 2,have
been found in [Ha-Li, 1928], We study J as an operator acting in LMO,a)
in 4.1 , as an operator acting in spaces of Holder continuous functions
in 4.2 , and we give a short account of smoothing properties of J in
other spaces of fractional order; in particular we consider Sobolev spaces
of fractional order. In 4.3 we give some compactness results for the
operator J and also for more general operators.
4.1. Continuity Properties of the Abel Operator in L^ Spaces
For functions f and g defined on IR we denote by f * g the integral
D.1.1) (f *g)(x) = J f(x-y)g(y)dy .
IR
If it is well-defined we call f *g the convolution of f and g -
see [Ru,1974;Ch VI], [Ti,l937], [Bra,1978].
The Abel operator J acts on a function f by effecting a
convolution of it with a power. In fact
D.1.2) Jau = x * u in @,a)
where
a-1
f'p-7—r f°r x > 0
r (a)
Xa(x) -{
V.
0 for x < 0
and 'u(x) = u (x) for 0 < x < a, 0 for x < 0 and x > a .

65
Theorem 4 .1 .1 : I (j u£Lp@,a), 0 < a < «., l<p<1/a,
and s = p/A-p(a-£)) with £>0 then
D.1.3) II Jau|| < =^- A+ ll") II u||
Ls@,a) ' l0" £ LP@,a)
Proof: The inequality D.1.3) is an immediate consequence of Young's
inequality for convolutions, as stated in the following theorem.
Theorem 4.1.2 {Voung'i inequality): let f £Lq(IR), g£LP(IR), wkeie
1<q< + <», 1 <p< «>, — + — >1 . Tken
D.1.4) || f *g|| < || f II II g II
IT(IR) IZMIR) LP(IR)
wheie
D.1.5)
1 = 1 +1
r p q
For the proof of this theorem,see the book [Ti,l937; Ch IV] -
[Ha-Li,l928] have proved that if p = 1 or p = — , £ cannot be omitted
in D.1.3). We shall show this in Remark 4.1.1 . Hence J is not contin-
1
uous as an operator from L @,a) to L @,a) or from L @,a) to
oo 1
L @,a). For 1 <p<— the inequality D.1.3) can be improved. In fact
the following theorem holds.
Theorem 4.1.3: I<5 0<a < + °° and 1 <p<— , the. following Inequality koldi,
D.1.6) II Jau II p < C(a,p) Hull
L1-ap@,a) L @'a)
wkeie C(a,p) It, a constant depending on a and p.
We omit the proof of this theorem. The interested reader can
consult £Ha-Li,1928] for a direct proof. It is not so straightforward as
Theorem 4.1.1, and some technical arguments are needed to prove
it. Another proof can be found by using the Marc inkiewicz-Zygmund
interpolation theorem, see [St,1970]. For completeness we quote [ON,1963]
who exhibits D.1.6) as a simple consequence of a more general theory
on convolution operators, ( see also [Ha-Ma-Se,1984] for further
consideration about inequalities of type 4.1.6) .

66
Remark 4.1.1: Ja is not continuous as an operator from
L @,a) to L ' ~a' @,a), even if a is finite. The following function
u is a counter-example.
1 (log 1) ~6 for 0 <x <i
D.1 .7)
We have u£L @,1) if 6>1, infact
1 ,, ., , 1-B
, 1
for x > -j
; u(x)dx = ii^i
0
But 1
Ja u <t L1 a @,1) if 1<B<2-a .
Indeed, for x < 1/2 we have
1 1 o-1 ■,,
Ta -, . 1 r x dA
J u(x) = jrj-v- J
r(a) i xA-XI-°(log-yB
xa-1(log 1)
> ——
(S-DIMa)
Hence
1 -1- 1 -2=1
J Ua G(x) I 1"a dx > 1 J 1 (log 1) 1"a dx ,
0 ' F-1)r(a) 0 x x
and this is equal to «> for 6 £ A,2-a). In fact -^- < 1 for these values
of 6 .
Remark 4.1.2: Ja is not a continuous operator from L ' @,1) to
L°°@,1). We prove that there exists a function v£L ' @,1) such that
Jav € L°°@,1 ) . Assume the contrary, namely Ja u eL°°@,1) for every
u £ L ' a @, 1) .BiBn. fori every tp £ L @, 1) we should have
1
D.1.8) Jju-ipdx<+«>.
' 0
From this inequality and from
1 a 1 1 1 a-1
J J u tpdx = — J (u(t) J ip(x) (x-t)u dx)dt
0 T(a) 0 t
we conclude that for every ip e L @,1) and for every u£L @,1) we have

67
D.1.9)
_J
T(a)
1
1
a-1
J (u(t) J tp(x)(x-t)u dx)dt < + c=
'0 t
which implies (see [Ro,1968; Ch. VII]
D.1.10
1
1 J <p(x) (x-t)a 1 dx £ L 1 a @,1]
T(a
for every ipfL @,1). However for ip(x) = uA-x), with u as in D.1.7),
the inclusion D . 1 . 10) ( see Remark 4 .1 . 1 ) does not hold,and we have arrived
at a contradiction.
1 a~ ~
Theorem 4.1.4: I <j p>- , u£Lp@,a), then Jau £ C P [0,a] and
u — — u —
D.1.11) || Jau IIro + a P [Jau ] 1 < a p C(a,p) II u||
LP@,a)
wheie C(a,p) -it, a constant tkat depend*, on a and p and the iem-tnoim
[u] -ii de^-ined ^on. 0 <-y < 1 by
tu] - sup '"'^-"'y'1
Y x,y£[0,a] |x_y|Y
Proof: Young's inequality for convolutions yields
1
D.1.12)
II J0 u|| <
r(a)f-2Erl) p
1 l|U II r>
1 -4 Lp@,a)
Now for y>x we have (using Holder's inequality with - + -, = 1
after the second sign < )
, -,a , , ,a , > , 1 , |u(t)|dt
IJ u(y) - J u(x) I < r^T J T^-
x(y-t)
r(o0 J
o '(y-tI-a (x-tI-a
|u(t) Idt
L_ ( ] dt
(a) I x (y-t)A-a)p'
1/P'
II u ||
LP@,a)

68
1
r(a)
J ( ] 1
0 V (x-tI"a (y-tI'0
P'
dt
1/P'
II u II
LP@,a)
Using the inequality (b-a)q < b^-a^, valid for b>a>0,q>1 we get
P'
(( ] ]
0\ (x-tI-° (y-tI
dt
x
< J
oMx-tI1"^' (y-t)A-a)p'
dt
1-A-a)p' 1-A-a)p' . ,1-A-a)p'
x y + (y~x)
1 - A-a)p'
Furthermore
Y
J
x (y-t
dt
A-a)p'
1/P'
a-
(y-x)
1 -
ap-l\
P-1/
—
P
1
P
Theorems 4.1.3 and 4.1.4 can be extended to the more general
operator A defined by
D.1.14)
(Aau) (x)
frit J
K(x,t)u(t)
r(a> 0 (x-tI
dt @ < x < a)
where K is a function on the triangle
2
T = { (x, t) £ m : 0 < t < x < a} .
The next two theorems generalize Theorem 4.1.3
Theorem 4.1.5: Let be K£L°°(T) -in the fioimula {4.1.14) define the opeia-
toK A . Then -ii. 1 <p<— .the oaeiatoi A -ii cont-inuoui Anom
a ° ^ a ' a u
LP@,a) to Lq@,a) with q = j^~ , and
D.1.15) || A u|| < C (a,p) II Kll II ull
Lq@,a) L (T) ]
wheie C.(a,p) -it, the iame comtant at, -in Theoiem 3.1.3

69
Theorem 4,1 .6: A -Li, a contA.nu.oui, opmatoi ^lom L @,a) to
_J
L ~a @,b) and ^lom L1'a@,a) to Lr@,b) &oi e.ve.iy e£ @,y^],r>1,
b£[0,a], (b<+«> xl<5 a = +°°) .
fu.itke.imo ie.
D.1.16 1) II A u II _J < C,(a,b,e) IIKII II u||
a T1-a £@,b) " L°°(T) L'@,a)
wke.ie. C9(a,b,e) a.& ike. iame. constant at, In D.1.3), and
D.1.16" ii) II A u II < C,(a,b,r) II KM Mull ,
0 Lr@,b) ^ L°°(T) L ' @,a) ,
1 /r 1
mknne. C2(a,b,r) = yj—y [1 + A-a)r] 1~a+r .
The proof of this theorem follows immediately by Theorem 4.1.3 .
For p>l/a,the following theorem generalizes Theorem 4.1.4 .
Theorem 4.1.7 I<J K £ CX (T) , thin A u £Cy([0,a]) ^01 u£LP@,a)
who.10. y = min {a- — , X] and p>1/a . Faitke.imoie. fioi X > 0-the. following
HiitX-mate. holdi:
1
a- — ,
D.1.17) MA u|| + ay[A u] <C(a,p)a P (IIKII + aA [K] )|| u II ,
Otoo OtjJ- °° A p
«fce*e C(a,P) - -^ (_El}) 1 ~ P .
We leave the proofs of the last three theorems to the reader as
exercises.
4.2..Continuity Properties of the Abel Operator in some
Spaces of Fractional Order
4.2.1. Holder Continuous Spaces
In this paragraph we give some results about the continuity of the
Abel operator acting in spaces of fractional order, for example in the
Holder spaces ca[0,a], 0< a <1 .
The following theorem and many other properties of the Abel operator in
Holder continuous spaces have been found by [Ha-Li,1928] .

70
Theorem 4.2.1 Suppoie
D.2.1 i) 0<a<1,0<B<1-a,
D.2.1ii) uec6[0,a] , u @) = 0 .
Than Jau £ Ca +B[0,a] and
D.2.2) [Jau]a+6 < C(a,6)[u]&
wkeie C(a,8) -it, a comtant de.pe.ndj.ng only on a and 6 .
Proof: For 0 < h < x < a we estimate the difference
D.2.3) Jau(x) - Jau(x-h) .
Using / u(x) (t-y)a-1dt = ^J*! (x-y)a for y=0 and y=h and
y a
x h x
splitting /=/+/ • we find
0 0 h
x -1 x -1
r(a) [jau(x)-Jau(x-h)] = / u(x-t)ta dt - / u (x-t) (t-h)a dt
0 h
= Hi20 [xa _ (x-hH] _ ; [u(x)-u(x-t)] ta_1 dt
x
/
h
- / [u(x)-u(x-t)] [ta 1 - (t-h)a 1]dt
We now denote by A,B,C the absolute values of the three terms of the
last sum and give upper estimates for each of them:
I u(x) r a . ,.an I 8 8 r a , K\ai
A = a [x - (x-h) ] < —— x [x -(x-h) ]
The concavity of the function x -» x for x>0, 0<y<1 (for 0 < r < s
we have sT - rT < (s-r)T) yields
x [x -(x-h) ] = [x -(x-h) ][x -(x-h) ] + (x-h) [x -(x-h) ] <
. a + 8 , , .B, a . . . a,
< h + (x-h) [x -(x-h) ] .
We distinguish the two cases x < 2h and x > 2h .
In the first case we have
(x-h) [x -(x-h) ] < h ,
in the second case we find (using the fact that the first derivative of
x is decreasing)
(x-h) [x - (x-h) ] < a (x-h) h<a h

71
In either case
(x-h) [x -(x-h) ] < h
therefore
2[u]
A < &- ha + 6
h
B = / [u(x)-u(x-t)]ta dt <
1 0 '
r , *? .a+B-1 ,,. [u]6 ,a + B
* [u]6 I t dt = ^1- h
C = I J [u(x)-u(x-t)] [ta_1 - (t-h)a-1] dt [ <
1 h l
x
h
< [u]R h J s Is - (s-1) Ids <
1
< [u]R ha + 6 J t6| ta'1 - (t-i )«-"" | dt .
1
Now from a + B-2<- 1 and a - 1 < 1, we conclude that
c (a,6) = J tB|ta_1 - (t-1)a_1Idt <+ ».
1
Therefore we have
C < c, (a,6)[u]6 ha + 6 .
Collecting the estimates for A,B,C, we obtain
|jau(x) - Jau(x-h)| < c(a,6)[u]6 ha+6
with
c(a,8) = r^yi| + ^ + cl(a,6)} .
Theorem 4.2.1 illustrates some analogies and differences between
integral operators of fractional and of entire order.
Indeed, it is known that u£C [0,a], 0<6<1, implies that the primitive
x 1+8
function Ju(x) = J u(t)dt belongs to C [0,a]and the repeated integral
0
J u, for n £ M belongs to C [0,a]. Analogously if u satisfies the
condition u@) = 0 and is in C [0,a], 0<6<1-a, then Jau £ ca+ [0,a] .

72
However, in contrast to the case of applying the operator J = J
(or one of the operators J with n £ W ) we require the additional
assumption u @) = 0 for 0<a < 1, if we want Jau to be "more regular" than u.
a
In fact for the function u s 1 we obtain (J u) (x) = r /X + 1* and we thus
have a simple example of a function u £C°°[0,a] transformed into a
function that is notevenin C1 [0,a]but only in Ca[0,a]. So, if u@) 4=0 there is
a loss of regularity. We remark that theorem 4.2.1 is not true if a+B =1.
See [Ha-Li,l928] for an illuminating counter-example.
4.2.2. Sobolev Fractional Spaces
In this section we shall give some continuity results for the
operator J in Sobolev spaces w"' @,a) of fractional order. To this pur-
0 P
pose we recall the definition of W ' @,a) for 1 < p < +°° , 0 < 0 < 1 . For
more details the reader may consult the monography of Adams A97 5) and
[Ku-al., 1977].
Definition 4.2.1-. Tot o < 0 < 1 and 1 < p < + °° we denote, by w" @,a) the
iabipace. o<5 faunct-Loni A.n L @,a) w-ith
1/p
D 2 4) lul = A 1 lu(x)-u(_t)_|f d d '
<4-2) lul0,P,(O,a) [iQiQ |x_t|1+P0
We nqulp thlb 6pacz with thu no/im || II n , . glvnn by
<4-2> "ull0,p,(o,a) = MuMLP@,a) + lul9,P,@,a) "
The space W'p@,a) is the completion of C [o,a] with respect to the
norm II II a ,_ ,, hence a Banach space.
y r P,( u, a)
Remark: For 0 a nonnegative integer we take as W the set of
functions u defined on @,a) which, together with all their derivatives up to
order 0, are in Lp@,a) .
For simplicity of notation we shall often write II u II _ instead
u, p
U|I0,P,(
ambiguity).
of Null n ... . and I u I n ^ instead of I u I n ,,. . (when there is no
0, PA 0, a) t),p o,p, (u, a)
Lemma 4.2.1: Let u £ W9'1 ( 0, a) , 0 < 0 < 1 . Then <joH evely q e [1 , j^q)
we have

73
[ -1-A-0) 1-1
D.2.6) Hull n < c@,q) a q I u I „ 1 + aq ||u II .
Lq@,a) L 0'n Ln@,a)
whe>ie c@,p) ■ih an app/iopi-tate. comtant.
For the proof see Appendix 4.A .
Theorem 4.2.2 I <J u e W0,1 @,a) , with. 0 £ [0,1-a] (tfo* 0 = o we take.
W°'1@,a) = L1@,a)) then
D.2.7) |jau|_ < c(a,0,e)ae(lul1 „ + a-0 Null )
u + a e, i i,a Li @;a)
whefie. c(a,0,e) -ca a corci-tarc-t that depending only on a,0 and £ .
Proof: (i) We first treat the case 0=0. Noting that
i«pie t = M dy J i«Pty)^x)i dx
0,1 o o (y-xI u
for any function ip £ L @,a) and that
v x
■n / > I ,0 / > Td , , , f I U (t) | dt , ( r 1
r(a) U u(y) - J u(x) | < J J— ' + J {
x (y-t) ' " 0 (x-tI a
(y-t)
1 } |u(t)|dt
for y > x,we get for 0<a'<a - £ <a, the estimate
r(a) Uau| , < 2A + 2B
CX t I
where
0 0 (y-x) a x (y-t)
B = J dy J ^^ J |u(t) I ( ^ - 1 -,.„) dt .
0 0 (y-x)' a 0 Mx-t) ' a (y-t)' a/
We treat A and B separately :
(a <**-<■ lu(t) I dx
a = ; dy ; dt ; — ' ^^
0 0 0 (y-tI a(y-x)' a

74
Since
i, J dy jf lu(t>' [ya: - <y-t)a'J dt
a 0 o (y-tI-a ya (y-tH'
< A J dy ; i»(t)idt
- a 0 0 (y-tI-a+a
J dy J -Lu(t)'dt , = J |u(t) Idt J ^ r
0 0 (y-tI-a+a 0 t (y-tI-a+a
a-a'
L @,a)
we obtain
a-a'
A < — II u II
a' (a-a') L @,a)
ay y
B = J dy J |u(t)|dt J
dx
0 0 t v (x-t) (y-t) (y-x)
= r 1-x dx . r dv r lu(t) ldt
j 1+ai aA ■ j ay j i-a+a'
0 a1 aA-X)' a 0 0 (y-t)' a a
- a-a'
< 2A +-L) ^ ||u II
a-a' L @,a)
Hence we have
■,a . 10a n n
a E' ' (a-e)er(a) l'(o,a)
Now we prove D.2.7) for 0£ @,1-a]. Observe first that
T" / \ I T01 , > T01 , > > r (U(t)-U(x)) ,.
T(a) (J u(y)-J u(x)) = J -5—— \_ dt
x (y-t) ' a
x y
+ \ (u(t)-u(x))[(y-t)a-1 - (x-t)a-1]dt+u(x)J ta_1dt
0 x
for x<y. We shall use the following notations:
A(x,y) . ) l(u(t)-u(x)lldt t
x (y-tI %

75
x
B(x,y) = J |u(t)-u(x)I[(y-t)a - (x-t)a n]dt ,
y n-1
C(x,y) = lu(x)I J ta ' dt
x
0' = 0 + a - e .
Then we see that
D.2.8) r(a)|JaulQ, 1 < J dy J A(*^>, dx + J dy J ^¾^
0 -1 o o (y-xI+0 o o (y-xI °
and
Now
/dy/ CU'y\%,
O O (y-x) ' u
lA(x,y) I < J l"<y>-"(*>■ dt + |u(y)-u(x) I ^-^
x (y-t)
Jdyjf dX1+0- J IU(y)-^)! dt- Jdyf dt ■"(yl-ujtH }
dx
0 (y-x),TU x (y-t)' " 0 0 (y-t)' " 0 (y-xI+0'
= J_ ? dv y( l"(y)-u(t) I (y0'-(y-t)9') .
0 0 (y-t) (y-t) y
< JL t dy T l"(y>-"(t)l < af |u|
- 0' 0 Y 0 (y-tI-a+0' S 0' 0'1 •
Furthermore
0 0 (y-x) 0 0 (y-x)
e
* IT |U|0,1
Therefore
D.2.9) J dy J A(x'y'^, < ae@'-1 + a'1) |u|Q
o o (y-x) '

76
Now
J ^ J BU'y\%, < J dy J ^ J |u(y)-u(t) I ( (x-t)a-1-(y-t)a-1)dt
0 (y-x)
0 (y-x)
+ J dy J l»(y)-»(x)ldx ; ((x-t)«-1_(y_t)«-1)dt
o o (y-x) o
a-1
ay y
< J dy J |u(y)-u(t)Idt J ((x-t)u '-(y-t)
0 0 t
a-1,
dx
(y-x
1+0'
1/dyf i»<y>-»(x)i dx
a 0 0 (y-xI+0 -a
1
< a£lu| (-1 + J (Xa-1-1) A-X)-1-0'dX)
,u Q
J (Xa-1-1) A-m-0'
1-a,,a-1
1/2 1
dX = ( J +/) (A-X1 ")AU ' A-X)
0 1/2
-1-0'
idX
c(a,9)
; dy ) B(x,y)dx|
0 (y-x) l+U
< ^ c(a,0)lul1fQ .
Therefore
a y
D.2.10)
0
Now consider the last term of sum D.2.8). We have
y n<„ ,,^ 1 a y
0 0
MyfS^, <i?dyf <l»tx)-u<yil + lu<y)
0 0 (y-x) ' ° a 0 0 (y-x) ' °
/ a a, ,
(y -x ) dx
0 0 (y-x) 0 0 (y-x)
y a_ a
1 1-Xa
a ii . 1 r I u (y) i , , i -a ,,
Tluli,e + «i >TdY /T7T,i+0' dX -
0 y^
0 A-X)
0 y

77
By Holder's inequality
a |u(y)Idy
J o_r i
o y
0-e
9-e
a o-£
J Y ° 2 dy
Lo
0-4
1
J lu(y) I
0
1+|-0
1+f-0
ay
20 2
< — a
- £
a 1+-|-0
J lu(y)I ^ dy
1+ 4-0
and Lemma 4.2.1 yields
1
a 1+4-0
J |u(y)I z dy
1+|-0
< C@,£)
£ 1-0
2 2
a |u| + a Null
0,1 L'(o,a)
Therefore
D.2.11) J dy J c<x'y) dx < c(a,0,£)
0 0 (y-x) ' B
a£|u|Q . + a£ 9 Hull .
a'' L'(o,a)
Finally,by D.2.9)-D.2.11) and D.2.8),
|J u| , < c(a,0,£)a
a 9 Null
L1@,a)
+ lu|
0,1
Theorem 4.2.3 I<J u£W0,1(O,a) {,on 0£ A-a,1) then Jaud W1,1@,a) and
D.2.12)
d ,a
-b— J u
dx
0+a-1
< c(a,0)a |u|Q . + a Mull _
-0,,
He.nn c(a,0) de.pe.ndi only on a and 0.
Proof: In order to demonstrate that Jau £W ' @,a) we show (see [Ku-Jo-
Fu,1977]), that there exists a sequence (v )cW ' @,a) such that(v )is a
Cauchy sequence in W ' @,a) and v -»Jau in L @,a).
1 0 1
Let (u ) be a sequence in C [0,a] such that u -»u in W ' @,a) and
consider v = J u . We have
n n
d 1 ( Un(x) X Un(x)-Un(t)
d - '--* - 1 ' n + A-a) J — ^r-r dt
v„(x)
dx nVA' r(a) V 1-a
0 (x-t
2-a
Now, by Lemma 4.2.1 we have (with £ = 0 - A-a)
fv
dx n
. < a£ ( lul. + a 0||uJ| 1 \
L1@,a) " V n 1,0 ^L1@,a);

78
1 1
and therefore (by Theorem 4.1.1) v £W ' @,a). For
n
we obtain
,,.(x, = _J ^u(x) ,, > ? u(x)-u(t) ,
*(x) T(a) V 1-a ( °° I , 4.,2-a dt
xx 0 (x-t)
II v'-iJjII , < c(a,0)aE(lu -ul. o+a~0 II un-u II .
n L'@,a) n l,W L'@,a)' '
II v -Jaul| < c(a)aa|| u -u 11 .
L'@,a) L @,a)
Furthermore -r— J u = 4> and
dx
II g| Jaull 1 < c(a,0,e)ae(|u| + a-0 II u II 1 ) .
L @,a) ' L @,a)
Before .closing this paragraph we remind the reader that there
exists a large literature on the properties of Abel operators in spaces
of fractional order. In [Ha-Li, 1928] the Abel operator is also studied
for functions u satisfying the "integrated Lipschitz conditions"
a ,
D.2.13) / |u(x)-u(x-h> |pdx = 0(hpK)
0
where 1<p<«>, 0<k<i . For these functions they show that if
0 < a < 1 - k
then
D.2.14) J I(Jau)(x) - (Jau)(x-h)|pdx = 0(hp(k+a))
0
(in D.2.13) and D.2.14) the righthand side can be replaced by 0).
In [K6,1974] inclusions of the type
ja(W0,P(O,a) ) cW0 + a_e'P(O,a) for 1 < p < + «. ,0 < 0 < 1 are found.
Other results and references can be found in [Bi,l983], [Bi,1984].
4.3. Compactness of Abel Operators
In this paragraph we present a few compactness properties of Abel
operators acting in spaces of Holder continuous functions and in L .
We first list some well known prerequisites from functional analysis.
For more details we refer to [Ru,1985], [Sm,1964], [Ta-La,1980].

79
A linear operator T : X~>Y, where X and Y are Banach spaces,
is called "compact" (or "completely continuous") if for the ball
B = {u £ X II ull < R} the image T(B) is relatively compact in Y, i.e.
the closure T(B) is compact in Y. We shall use the important subsequence
property of the notion of compactness: -in a Banack ipacQ V a iubiQt U
-ii HQlat-ivQly compact -i^ QVQfiy iQquQncQ 0(J QlQmQnti (un) In U conta-ini
a iubiQquo.n<^e. that convQUQQi -in V. We shall use the following
Theorem 4.3.1 (Arzela-Ascoli): LQt J be a iubiQt o& C [a,b] , with
- «> < a < b < + oo. Th.Qn J -ii HQlat-ivQly compact Jin C [a,b] -i^ and only -i^
all ^unct-ioni -in J alQ Qqu-iboundQd and Qqu-icont-inuoui .
By saying "all functions in J are equicontinuous and equibounded"
we mean that there exists a real number M such that II u|| „Or , , < M ,
for every u £ J (equibounded) , and that for every £ > 0 there exists a
6 > 0, depending on £ only, such that Ix-y| <6 implies |u(x)-u(y)l<£ ,
for every u £ J(equicontinuity) .
The following lemma gives a useful example of a compact set in
C X [0,a] , 0 < X < 1 .
Lemma 4.3.1: LQt J be a iubiQt o $ Cy[o,a], wfie-te 0 < y < 1 . I <5 all ^anct-iom
■in J aHQ Qqu-iboundQd and
D.3.1) [u] < L < + oo $01 QVQiy u£J
th.Qn th.Q iQt J -ii IQlat-ivQly compact -in C [0,a] ^oK QVQly X £ @,y) .
Proof: Let (u ) be a sequence in J. From the equiboundeness of J and
n n ^ ^
from condition D.3.1), assuring the equicontinuity, we conclude that
there exists a subsequence (u ), of (u ) and a function u £C [0,a]
^ n ' k n n
such that (u ) converges to u in C°[0,a]; by D.3.1) u£Cy[0,a] . Now,
to prove that J is relatively compact in C [0,a] we show that
Cv[0,a]c=c [0,a] for 0<X<y and that (u ) converges to u in C [0,a].
nk
For v £ Cy[o,a] and 6 > 0 we have
[v] &V~X for |x-yl < 6
y
D.3.2) lv(x)-v(y)l <
!x-y|X
-X
2II v II 6 for I x-y I >
C°[0,a]
Minimizing the right hand side with respect to 6 gives

80
1 -A 1 -A K
D.3.3) [v], < 2 y II vll y ([v] )y .
X ' C°[0,a] y
By D.3.3) CX[0,a] =>Cy[0,a] , hence u £ CX [0 ,a]
and X^ ,, A^
[u - u], < 2 Ly Mu, - u II y
nk X ~ nk C°[0,a]
Therefore since (u ), converges to u in C [0,a],*the sequence (u ),
nk k nk k
converges to u in C [0,a].
Now we present some compactness theorems for the Abel operator
x
a l0" 0 (x-t) a
Theorem 4.3.2: Let be K£C (T) w^-Cfe 0<X<1 and p>— . Tfien -the ope^ta^o-t
Aa : LP@,a) - Cy'([0,a])
-ih compact fion. cvciy y'£[o,y), wfie-te y = min {a - -,U .
Proof: Let be X £ @,1) and BR = {u £ LP(o,a) | II ull< R}. To prove that
A (Bn) is relatively compact in Cy ([ ,a]) for every y" <y ,
CX K
we use Lemma 4.3.1. If u £B_ we have, by Theorem 4.1.7
K
a-1
II A ull + ay[A u] < C(a,p)a P (II KII + aA[K],)R .
CX oo cx y — °° A
y '
Therefore by Lemma 4.3.1, the set A (B_) is relatively compact in C ([0,a];
for every y' < y .
Let be X = 0. By some manipulations similar to those made in the
proof of Theorem 4.1.3 we get for every u £ B_
D.3.4) I (Aau) (y)-(Aau) (x) I < j^j (J2^1) ( II K 11^ (y-x)
+ sup|K(y,t)-K(x,t)I.aa_1/p)R.
t
Since K £C°(T)f D.3.4) gives the equicontinuity of{A u}, u in C ([0,a]).
The proof is completed.
Now remember T = {(x,t)| 0<t<x<a} .
Theorem 4.3.3: Let be K £ C° (T) and 1 <p<—. Thin A ii, compact at, an
_ r - a a '
ope.ia.toi fiiom Lp@,a) to Lq@,a) (Jet eveiy q£ [ 1 , -,^.-)

81
The proof of this theorem (which we omit) is essentially analogous to
that of Theorem 4.3.2, ©xe&pt for some technical arguments .
The interested readers is referred to [Kr-al, 1976 ; ch.2,§3,Th .8 .1 ] ,
see also [Mi,1970] .
Remark 4.3.1: Theorem 4.3.3 is not valid for q = " even if
1 1
I,—) (for p = 1 and p = — , we have
ing Theorem 4.1.3),that the inclusion
p£ A,-) (for p = 1 and p = — , we have seen (see the Remarks follow-
J (L^) cL " is not valid). We show this by an example due to
[Kr-al,1976]. Let
n p for 0 < x < —
Un(x) .' ,
for — < x < 1
n -
where p£ A,-). We have
J u (x)
n
n1/P xa , n 1
1/P
[xa-(x --)a] for I < x < 1 .
rA+a) L v n
Therefore
l|unN p = 1 '
n LP(Q,D
and --a
D.3.5) I|jau-Jaul| D > A-°P)P [1-(^I/P]
n m -^2— rA+a)
L1-aP@,1)
for m>n. We see that (J u ) cannot contain a convergent subsequence
. P n n
in L1-°P @,1).
Appendix 4A; proof of a Lemma
In this appendix we prove Lemma 4.2.1. We recall a lemma of [Jo-Ni,196l]
(see also [Ku-Jo-Fu,1977;p.2351 ) .
Lemma 4 .A. 1 : Let u£L (C,a) and iuppoie that the. following condition
-L& iat-L&fa-led
ai - i P
K (u): = sup Z (a.-a^.,)' P(J |u(x) ~ _a— J u(y)dyldx) < + °°
1=1 a.^ i i-1 a.^
wheie p > 1 and the. iuptemam li, ovel all iabdlvliloni

82
0 = a <a1<a?<...<a = 1 o ( [0,a] ,n Kunn-ing thfiougk all poiA.tA.ve -in-
te.ge.16. Then
1 a
u(x) -^/ u(y)dy G L@,a)
a 0
faon. eveiy g£ [1,p) and
1 1
, a J. _ J. 1/p
II u - j- J u(y)dy|| < c(p,g)aq p K (u)
a 0 Lq@,a) P
0 1
Proof of Lemma 4.2.1: For ufS @,a) we have
a. a. 1
n -■> 1 1 T^0
K 1 (u) < sup I (ai-ai_1) ((ai-ai_1) J J I u (x) -u (y) I dy dx)
T=Q 1=l ai-1 ai-1
1
n i i i < > < > ij 1-Q
*- / r j r u (x) -u (y) dy.
< sup Z (J dx J -—^-^— 'jl' —-)
1=1 ai-1 ai-1 lx-y!
a. a. _1_ _J_
< sup ( Z J" dx J" |u(x'^y" dyI-0<|u| 1-8 -
1=1 ai-1 ai-1 lx_y!
Therefore
1-0
K_i_ (u) < |u|1,0 '
1-0
and Lemma 4.A1 yields D.2.6).

Chapter 5: Existence and Uniqueness
Theorems
In this chapter we shall prove some theorems on existence and u-
niqueness for general linear and nonlinear Abel equations. We treat the
linear equation
1 X K(x,t)u(t) ,. ,,.
J i-„ dt = f(x) , 0 < x < a ,
r@) 0 (x-tI"a
in 5.1, the nonlinear equation
J K(x't'"lt)) dt = f(x) 0<x<a
r(a) 0 (x-tI-a
in 5.2. We mainly use classical arguments, such as the successive
approximation method. In short, we refine well known existence and uniqueness
results for Abel equations, see [At,1974] , [Ko,1930] , [Me,1976] ,
[Ta,1930] , [Tr,1957] , [To,1923] .
5.1. The Linear Case
In the following theorems we use the notations
C*[0,a] = {u £C6[0,a]|u@) = 0} for 0<6<1 ,
D«f = _i <L x f(t)dt for 0 < a < 1 .
r<1-a) dx J, (x_t)a
Theorem 5.1.1: The. equation
E.1.1) 1 x u(t)dt *, i „
rT^T * 1-a = £(x)' °<x^a '
l0" 0 (x-t) ' a
kai, a unique solution u£C*[0,a] ui-Lth. 0<8<1-a -i^ and only 1(,
f JC°+ [0,a] . fun.tko.nmofin u £ Daf and
E.1.2) II u II 6 < C(a,6) A+a6) [f] .
C [0,a]
wkeie C(a,8) -it, a constant that de.pe.ndt, on a and 8 only .
We give only a sketch of the proof. The uniqueness of the solution
follows directly from the equation, see also [To,1928],

84
In fact, posing f =0 in E.1.1) we have
x 1 _
J u(t)dt = J Jau(x) = 0, that is usO in [0,a] .
0
Furthermore if there exists a solution u£C*[0,a] of E.1.1), we have by
Theorem 4.2.1 ,
f = Aecf6 [0,a] .
Extending feC* [0,a] by setting f(x) = 0 for -°° <x<0 we
have u = D f with
E.1.3) Daf(x) = rA°a) J (f(x) - f(t))(x-t)_a-1 dt .
—oo
To convince yourself of E.1.3) you may accept the following formal
argument (in which integration by parts is used)
naf = A 1 X f(t)dt = 1 x f'(t)dt
dx T(i-a) J . ...a TA-a) J , .-a
-oo (x-t) -oo (x-t)
= ] "r A. (f(t)-f(x)) dt = —2 *( f(x)-f(t) dt
TA-a) J dt um r(x" , ,.,a rA-a) J , .-1+a at '
-oo (X-t) -oo (X-t)
For a non-formal proof see [Ha-Li,1928,Th.19].
Inequality E.1.2) can be proved by an argument analogous to that
used in the proof of Theorem 4.2.1 .
By using results of [Ha-Li,1928],similar theorems of existence and
uniqueness with estimates of type E.1.2) in subspaces of L^
satisfying the "integrated Lipschitz conditions" (we have defined this
space in 4.2) can be obtained. For example, Hardy and Littlewood prove
that if 1<p<+oo,a<8<1 and
a .
E.1.4) J If(x)-f(x-h)|pdx = 0(hp )
0
then the solution u = D f of the equation J u - f exists, u £LM0,a)
and
J |u(x)-u(x-h)|pdx = 0(hpF-a)) .
0
Many authors have studied the question of existence of solutions
for the Abel equation when the data function f lies in Sobolev spaces
W0,p of fractional order, see for example [Bi,1984], [Ko,1974]. Their
results are direct consequences of the smoothing properties of the Abel
. ,a . „0,p
operator J in W r .

85
For example recalling Theorem 4.2.3, we have the following result.
Theorem 5.1.2: I <J few"' @,a) jot 0£(a,1) then theie ex.-ii.ti, a
iolut-ion u {that -it, unique, i>ee Theoiem 1.2.1) -in L @,a) o<5 the Abel
-integral equat-ion J u = f, and we have
II u|| < c(a,0)a0-a(|f I , + a_9|| f II
L @,a) U' L'@,a)
Because a function absolutely continuous in [0,a] 'is in W ' this
Theorem generalizes Tonelli's Theorem 1.2.1 .
Theorem 5.1.2: Let be g-iven a Banach i>pace X and a bounded l-ineal
opeiato/i A : X->X. Xititume the iet-iei, f Anf to be notm-conveigent -in
n=o
X fioi eveiy f ex. Then, jo-t any g-iven gex, the equat-ion
E.1.5) (I-A)u = g ,
wheie I -ii, the -ident-ity opeiaton. ,-ii un-iquety iolvable -in X, the iolut-ion
be-ing g-iven by
00
E.1 .6) u = Z Ang .
n=0
Proof: (a) Uniqueness: Assume E.1.5) to have two solutions u.,u,.
For u = u^-u1 we have u = Au, hence u = A u for every n £ M . From the
oo
convergence of the series Z A u we now conclude that lim A u = 0,
n=0
hence u = 0 and u. = u~ .
oo
(b) Existence: u = Z Ang solves E.1.5) because
n=0
N
(I-A)u = lim (I-A) Z Ang = lim (g-AN+1g) = g .
N-»= n=0 N-»=
Remark 5.1.1: Observe that if the series Z An converges
n=0
in the operator norm, then
oo
E.1.7) || u||Y < || Z An|| . ,Y. • ||f II .
X - n=Q MX) X
By using the previous theorem we can prove an existence and
uniqueness result for the equation A u = f where

86
i c -, q * it, <. i > 1 r K(x,t)v(t)dt
E.1.8) (A v) (x) = rw- J ' ' , o<x<a .
a l Va' 0 (x-t) a
We consider this equation under the following conditions on K:
E.1.9 i) KEC°(T), whefie T = { (x,t) SIR2|0<t<x<a} ,
E.1.9 ii) K(x,x) = 1 $01 eveiy x£ [0,a],
E.1 .9 iii) |^ £ L°°(T) .
dx
Theorem 5.1.3: I <J K £ C° (T) iatli^lei, E.1.9 i) , E.1.9 ii) ,
E.1.9 iii) and f -ii a fiuyLct-Lon m-ith
E.1.10) Daf £ LP@,a)
then the equation
E.1.11) 1 ^ «<*'*>"<*> dt = f(x), 0<x<a,
1 l0" 0 (x-t) ' a
hai, a un-Lque iolut-Lon u£L @,a). Th-ii iolut-ion iat-ii^-iet,
E.1.12) Hull < C(Ma,p) II Daf II
LP@,a) LP@,a)
I 3K I
wheie M = sup tt- and C(Ma,p) -it, a constant depending iolely on the.
ij, I ox I
■indicated an.gume.nti.
Proof: Assume K to satisfy E.1.9 ii). Then
E.1.13) Aa = Ja(I-Ba)
where I is the identity operator in LP@,a) and B is defined by
E.1.14) (Bv)(x) = - ^-^ ;{v(t) | 3 (H(g,t) \ _d^} dt
a ^ o t dt> MC-t) ' a/ (x-C)a
with H(x,t) = K(x,t) - K(t,t) . Formula E.1.13) is a consequence of the
following calculation.
/» w i 1 Xr u(t)dt 1 x H(x,t) ,4.1.,4.
(Aau) (x) = TM I 1-a TT^Y ' 1=a" u^dt
a M0" 0 (x-t) ' a l (a' 0 (x-t) ' a
Ta , 1 r H (x, t) ... ,.
= J u + rT^T ' 1-a u(t>dt ■
1 l0" 0 (x-t) ' a
Applying the operator D we obtain, after a change of the order of inte-

87
gration,
,^°^ <. i <. i > , sin na r , ... r 3 / H(C,t) \ d? ■, ,.
(D A )u(x) = u(x) + — \ {u(t) \ ^ '-' _a — } dt ,
a n 0 t 3S V (C-tI a / (x-C)a
hence E.1.13). Now we see that E.1.11) is equivalent to the equation
E.1.15) (I-B )u = Daf .
Putting
E.1.16) L(x,t) = - SAlLJia ; 3 f"^'^ ) —^
n t di= MC-t) ' a/ (x-C
, 0(
we can use Theorem 5.1.2 for solving equation E.1.15). we must analyze
the convergence of the series
E.1.17) u = Z u
n=0 n
where u = D f and u = Bn u for n 7 IN .
o n a o
Noting the recursion
E.1.18) u = B u , for n £ IN
n a n-1
and considering E.1.16) we first give an estimate for |L(x,t)|. From
x -1
L(x,t) = - S1" na J" (x-C) a(C-t)a n{(a-1)(C-t) 'H(Ct)
t
+ H?(?,t)} d?
and
i(a-1)(C-t) 1H(C,t) + H?U,t)| < 2 M
which follows from D.1.9 iii), we get
x ,
|L(x,t) I < 2 M Sln TTa J" (x-C) a(C-t)a ' d? = 2 M,
t
the integral here representing the beta function which already appeared
in Chapter 1 (see also [Ab-St,1972]). Thus from E.1.16) and
x
(B v) (x) = \ L(x,t)v(t)dt we obtain
0
x
E.1.19) I(B v)(x)| < 2 M / lv(t)| dt for v£LP@,a) .
0
The convergence of series E.1.17) and, therefore, the validity of
Theorem 5.1.3, now, is an immediate consequence of the following lemma.
Lemma 5.1.1: Von 0<x<a and n £ IN we. have.

88
E.1.20) HuJI™ < c (p) B Mx)n llu II
n LP@,x) n °LP@,x)
with c (p) = p-n/p(n!)-1/p a.£ 1 < p < + oc , c («>) = 1/n! .
11 1
Proof: With — + — = 1 (in particular — = 0 and q = 1 if p ==°)
we obtain for v£Lp@,a) and 0<t<x<a, by E.1.19) and the H51der
inequality, 1
1--
E.1.21) I (B v) (t) I < 2 M II vll t p .
0 LP(Ort)
Insertion of v = u yields,
Hull < 2 M II u || p-1/p x for 1<p<«
LP@,x) LP@,x)
II u. II < 2 M II u l| - x for p = oc .
1 L°°@,x) " ° L°°@,x)
These inequalities exhibit E.1.20) as valid for n = 1. Assuming validity
for an index n = m> 1 we obtain by E.1 .21) 1
1--
|u (t) | < 2 M Null t p for 0 < t < x < a ,
m+' m LP@,t)
llu +1|l < BM)m+1 c (p) II u II - II w 11
m+1 LP@,x) " m ° LP@,x) LP@,x)
1-1
with w(t) = tm t p and
m+1
II w 11 = T-j— in the case 1 < p < o=> .
LP@,x) (p(m+1))I/P
In the case p = » we use E.1.19) with v(t) < II v II
1 ' L°°@,t)
We obtain, considering the definition of the c (p) the desired result
II u || < B Mx) m+1 c, (p) II u^ll
m+1 LP@,x) " m+1 ° LP@rx)
The Lemma being proved, we see that the unique solution of E.1.11)
is given by E.1.17) and we can identify the constant C(Ma,p) in formula
E.1.12) of Theorem 5.1.3 . Formula E.1.20) allows us to take
C(Ma,p) = Z c (p) B Ma)n .
n=0
In particular we have C(Ma,o=>) = exp B Ma) .

89
The proof of Theorem 5.1.3 is completed. The condition E.1.4)
implies Daf £Lp@,a) for 1 < p < + «>. On the other hand, if p = + «° , f @) = 0
and f £Lr@,a), r > j^ , then Daf £L°°@,a) .
In fact, an integration by parts yields
naf - 1 "r £' (fc)dt - t1-0 f-
r A-a) ' oT ~ '
u a' 0 (x-t)a 1
and from Theorem 5.1.4 we find Daf £ C r [0,a] , hence D f £L°°@,a) .
Existence and uniqueness results can be proved also in spaces
C [0,1], The proof of the following theorem is analogous to that of
Theorem 5.1.3, therefore we only sketch it.
Theorem 5.1 .4: Let he K £ Cm+ [0,a] w-ith T a.6 -in E.1.9 i) and aaume
E.1.22) K(x,x) = 1 ion. eveiy x e [0,a].
let f be a function defined on [0ra] w-ith
E.1.23) Daf £Cm([Ora]).
Then the equation
i \ 1 r K(x,t)u(t) ,. c , . _
Aau(x) = TT^rr -f 1-a dt = f(x)' °<x<a'
a l l°" 0 (x-t) ' a
hat, a unique iolut-ion u -in C [0,a] and
E.1 .24) Null < C(M ^.,3) II Daf II m
Cm([3,a]) " m+1 Cm([0,a])
whete M - = II KII , and C(M , ,a) -it a constant depending
m+1 Cm+1[0,a] m+1
on M . and a .
m+1
Sketch of the proof: Uniqueness is guaranteed by Theorem 5.1.3.
For m = 0 the theorem follows from the fact that C [0,a] is a closed
subspace of L°°@,a) and from the validity of u = BnDaf £C°[0,a] for
every n £ ]N .
For m > 1 we can observe that
II u II v, < C(M ,a) II u .. II . . for 1 <h <n
n Ch[0,a] ~ m+1 n-1 Ch-1[0,a] ~ ~
where the constant C(M ,.,a) depends on M . and a only.
m+1 c m+1
Therefore
E.1.25) II u II < OH,, .,a) Hu„ II for n > m
n „mr_ i - m+1 n-m „or^ i "~
C [0,a] C [0,a]

90
and
E.1.26) Hull m < C(M ^.,a) l|Daf|l f or 0 < n < m .
n Cm[0,a] " m+1 Cm[0,a] " "
Furthermore, applying the inequalities E.1.20) and E.1.24), we obtain,
for n > m, n-m
BM.a|
E.1.27) II u || < C(M .,a) -. L—; 1| Daf|| n
n Cm[Ora] " m+1 (n~m)! C°[Ora]
In view of E.1.26) and E.1.27) we conclude that the series
E.1.28) Z u
n=0
converges in C [0,a] , its sum u solves the equation A u = f, and
II u || m < CW ,,a) II Daf II m
„mr_ , - m+1 ^™r- ■,
C [0,a] C [0,a]
We end the consideration of the linear case by reporting another
existence and uniqueness result which can be useful for explaining the
relation between the asymptotic behaviour of the data function f and the
solution u in the origin. The proof (we omit it) can be carried out with
the technique already used in the proof of Theorems 5.1.2 and 5.1.4
(compare [At,1974]) . In the theorem we are going to formulate there occurs
the space
C[0,a] = U {xYip(x) I ip£C°[0,a]}
Y>-1
Theorem 5.1.5: iil^tk ? £ Cm+ [0ra] (Jo-t an -integen. m>0 let the jane-
tlon f be Of) the faoum
f(x) = x f(x), 0<x<a,
and aaume
y>0, 0<a<1, yA-a) + 6 > 0,
K£Cm+2(T) , K(x,x) = 1 for 0<x<a .
Thin the -Integral equat-ion
—L ; «<*'*>"<*> dt=f(x), 0<x<a,
F(a) 6 (xu-tV~a
ha.i a unique bolut-lon u£C[0,a] and th-it, iolut-ion can be wi-itten -en tke
{elm
. . yA-a)+6-1 ~, .
u (x) = xH u(x)
wheie u(x) = b+xg(x) with g£Cm[0,a] and a conitant b .

91
We have. b=0 l{> and only l£ ?@) = 0. With a iultable constant c,
independent Of) ?, the estimate
Hull < ell f II m4,
Cm[0,a] " Cm+1[0,a]
-tA valid.
5.2. A Nonlinear Abel Integral Equation
For reasons of completeness, we now investigate a nonlinear Abel
equation for existence and uniqueness of its solution. This equation is
of the general type (see also [Bra,1978] , [Me,1976\])
E.2.1) 1 jf K(x,t u(t)) dt = f(x)> 0<x<a,
1 l0" 0 (x-t) ' a
where K : T x IR -> IR and f : [0ra] -» IR are given functions and
u : [0ra] -+ IR is unknown. The reader should recall the definition of T,
namely T = { (x, t) I 0 < t < x < a} . as usual, 0<a<1 .
In the following theorem we shall need three hypotheses on the
function K(x,t,w), namely
E.2.2 i) K £ C1 (Tx IR ) .
E.2.2 ii) Theie exlbtb a conitant M < + °° tuch that
11^ (x,t,w) - -1^ (x,t,w)l < Mlw-wl ^on. eveny
dX oX —
(x,t) fT and eveny w,w£IR.
E.2.2 iii) j- (x,x,w) > c > 0 (on. (x,w)e[0,a]xiR with a. constant c.
Theorem 5.2.1 : Aiiume E.2.2 i) , E.2.2 ii) , E.2.2 iii) (ul(llled
and let f &atl&(y the condition
E.2.3) J1~af £C1[0,a] , J1_af@) = 0 .
Then the equation E.2.1) hai, a. unique continuous solution. Tunthenmone,
Id (£.2.3) li iatli(led with f = f. and i
ipondlng iolutloni o ( E.2.1) the estimate
l( (£.2.3) It, hatlh(led with t - t and f = f2 and u. ,u2 ale the conne
E.2.4) || u..-u,l| < C. (a)exp{C0(a)Ma} II Daf1-Daf0ll
1 2 L°°@,a) ~ 1 2 1 2 L°°@,a)
holdi with conitanti C. (a) ,C? (a) not depending on t.,Z-. .

92
Proof: Applying to E.2.1) the operator J we obtain the
equivalent equation
x
E.2.5) \ H(x,t,u(t))dt = J af(x)
0
where
E.2.6) H(x,t,w) = ^-Z° ; K(y,t,w)dy , for (x,t) e T, w £ IR .
n t (x-y)a(y-t)' a
This function has the following three properties.
E.2.7 i) H e C1 (T x ir) .
E.2.7 ii) tt- -Li L-ipich-itz-cont-Lnuoui w-lth fie.ipe.ct to w
dX
w-ith the. iame L-ipich-itz conitant a.i tt— -in E.2.2 ii).
dX
E.2.7 iii) T7- (x,x,w) > c(a) >0 wkene. c(a) -ii a conitant depending
d W "~
iolely on a .
E.2.7 i) and E.2.7 iii) can easily be proved by standard analytic
methods from the parametric integral representation
1 1
E.2.8) H(x,t,C) = sin na J" -^-^ - K(X(x-t) +t,t,?)dX .
n OX1 aA-X)a
Differentiation of both sides of E.2.5) yields the integral equation
(with H = |^ for short)
X dX
E.2.9) H(x,x,u(x)) + j H (x,t,u(t))dt = Daf(x)
0 X
which is equivalent to equation E.2.5). If u is a continuous solution
of E.2.5) we obtain E.2.9) by differentiation.
Conversely if u is a continuous solution of E.2.9) we get by
integration
J H(x,t,u(t))dt - J1-af(x) = 0 ,
0
1 -a
using the continuity of H and the assumption J f@) = 0 .
Now Dini's theorems on implicit functions yields the existence of
a function ip with the following three properties
E.2.10 i) ip£C1([0,a]xi), where I is the image of [0,a] x m under
the function H(x,x,w).
E.2.10 ii) For every x £ [0,a] and z £ I we have
H(x,x,g) = z if and only if £ = ip(x,z).

93
E.2.10 iii) 0 < ip (x, z) < —-.—r- for every x £ [0,a] , z e I.
We prove only E.2.10 iii). By E.2.10 ii) we have H(x,x,tp(x,z)) =
and by differentiation we obtain
Hw(x,x,ip(x,z) ) tpz(x,z) = 1
from which, using E.2.7 iii), we get E.2.10 iii).
We see that equation E.2.9) is equivalent to
x
E.2.11) u(x) = (p(x,Daf(x) - J" H (x,t,u(t))dt) .
0 x
and this equation can happily be solved by the method of successive
approximations according to
uQ(x) = 0 ,
x
un(x) = tp(x,Daf(x)-J Hx(x,t,un_1 (t) )dt) for n E I .
We show that this sequence of functions converges to a continuous
function u solving E.2.11). For n> 1 we have
x
u ,.. (x) - u (x) I < —I—r If (H (x,t,u (t) )-H (x,t,u . (t) ) )dt| <
n+1 n - c(a) ' x ' ' n x n-1
0
x
< —A- I |u (t) - u , (t) |dt
- c(a) i nv n-1 '
where M is the Lipschitz constant of H (x,t,w) which is the same as that
of K . Therefore by recursion
n
E.2.12) |un+1(x) - un(x)l < jjL (^¾.) ||U1II
Li (O, a)
for every n> 0, and we recognize the convergence of the sequence (u )
to a continuous function u. The continuity properties of ip and H qualify
u as a solution of the integral equation.
To obtain uniqueness and the estimate E.2.4) we consider functions
f1 and f5 satisfying E.2.3) and solutions u1 and u? of E.2.1) for
f = f. and f = f„ respectively. Then
|Ul(x) - u2(x)| < ^y {|Daf1(x) - Daf2(x)|
x
+ M / |u.(t)-u,(t)Idt} ,
0 '
therefore
IV*)-u2(x)l ^ til Daf1-Daf2ll +
Li \U r 3.)

94
x
+ M \ |u (t)-u,(t)Idt} .
0 '
By Gronwall's inequality (see Remark 5.2.1) we obtain the estimate
Daf1-Daf0i|
1 2 L~@,a)
which implies uniqueness and E.2.4).
Remark 5.2.1: In the preceding proof we have applied the integral
form of
Gronwall' s lemma: Let u and v be cont-inuoui non-negative ftUnctloni on
[0,a] and let c be a nan-negative neal nambei. Aaume ^on. 0<x<a,
the -inequality
x
v(x) < c + \ u(t)v(t)dt .
0
Then we have the estimate
x
v(x) < c exp (/ u(t)dt) ^on. 0<x<a.
0
MX
c(a)
|u1(x)
-u2(x) I
c(a)
For a proof of this Lemma see [Mi,1970].

Chapter 6: Relations between Abel Transform and
other Integral Transforms
In this chapter we study the relations between Abel and Fourier, Mellin,
Hankel and Radon transforms. We cannot achieve completeness, but we
indicate the most relevant and common relations. Sometimes, e.g. for the
Fourier and Mellin transforms, we study in more detail the relations
and we give also applications in the study of questions of existence,
uniqueness and stability, we begin with some relevant relations in the set
of Abel operators J (the fractional integral operators).
6.1, Relations of Abel Operators with Abel Operators
From basic real analysis we know that for a given positive number
c there is exactly one continuous and monotonic function ip : IR -» IR
obeying
F.1.1 i) ip(a + 6)= ip(a) - ¢F) ,
F. 1 . 1 ii) ¢A) = c .
This function is called the exponent-Lai fcunct-Lon with base c. We write
ip(a) = c because ip is the natural extension to real numbers as exponent
of the power c with an integer n.
Analogously, if A is a linear operator in a normed vector space X
o
one defines the powers A for natural numbers n recursively as A = I,
A = AA , and is then confronted with the question: Is it possible to
define a family of linear operators (A ) extending in a natural fash-
2 ^ ' a>o
ion the powers: A ?
In other words: Does there exist a family (Aa) of linear operators
with the following three properties
[on. all a,6 > 0,
F.
F.
F.
.1.2
.1.2
.1.2
i)
ii)
iii)
„a+6 a 6
A = A A
A1 = A ,
the map a -»
the map a-> A -ii aontintiou.& In a benhe to be ipec-l^-led.
tors A that satisfy these three p
are called fractional powers of A with exponent a
The operators A0 that satisfy these three properties (if they exist)

96
Evidently these considerations are still very different from a
precise definition and, as we can imagine, a precise characterization
of fractional powers of an operator is not as simple as the definition
of the exponential function.The reader interested in this type of
questions should see [Ba,1960]r [Fa,1983], [Yo,1965].
We briefly discuss the properties F.1.2 i) , F.1.2 ii) , F.1.2 iii)
for the Abel operator J acting in spaces of functions defined in [0,a],
especially in the space L [0,al. For more general and complete results
see [Bo,1930], [Ge-Sh,1964], [McB,1979] , [Ro,1975], [Ta,1930].
Consider the operator J for an arbitrary positive number a. We
formally define D by
F.1.3) Dau := Dn Jn-au for a > 0, n - 1 < a<n,
which is meaningful if Jn f has an n-th derivative, we use this
definition in Theorem 6 .1 . ~l . Here n is a natural number.
Clearly J = J, the ordinary integration operator. In the following
theorem (see [Ta,1930]) essential properties of Ja in C [0,a] are
established.
Theorem 6.1.1: Van. u £ C [0,a] we have.
F.1.4) lim Jau(x) = u(x) ^on all xe @,a]
a-»0 +
and
F.1.5) J6 Jau = Ja + 6u ^on. all x? @,a].
Remark 6.1.1: The statement
lim Jau@) = u@)
a->0+
is true only if u@) = 0. If, e.g., u ■ 1 then
Jau(x) = „ .1 ., xa -» 1 for all x > 0, but -»0 for x = 0 .
r(a+1)
Proof of the theorem: Fixing x>0 and integrating the expression
x a-? x
/ t ^ \ u(x)dx dt
0 x-t
by parts, we obtain
Jau(x) = yj^j J" (x-t)a~1u(t)dt
= pt-T {x°i u(t)dt + A-a) J" (ta-2 \ u(T)dT)dt}
va' 0 0 x-t

97
and from lim V (a) = <*> we deduce
a-»0 +
xa x
lim ^7777 J" u(t)dt = 0 .
a_,0+ K ' 0
1 x
Defining £ (t) = -r \ u(-r)dx for 0<t<x,
X x-t
£x@) = U(X), £x(t) = Ex(t)-U(x) ,
we see that \1 (t) | < 2 Hull for all t e [0rx]
X C°[0,a]
and that for every fixed positive r\ there exists a positive number 6 such
that
| £x(t) I < n for 0 < t < 6 .
By splitting we find
1 —a ,- ,.a~2 r , ., . .. 1-a , . a
Yj- J (t J u(T)di)dt = u(x)x
1 va' 0 x-t rA+a)
'A+a)
6 x
1-a r .a-1 - ... ,. 1-a f - ... .a-1 ,.
+ r-j—x J t £ (t)dt + vi—r J £ (t)t dt .
r (a) JQ x r (a) ^ x
We have lim -=-Ti r u(x)x = u(x) , and the above bounds for £„(t)
a_0+ l A+a) x
imply
and
llrfr J £ (t)ta 1dt| < Trj^-y 2|| u|| (xa-6a) -+0 for a - 0-
i (a) 6 x l ( i+a) c°[Ora]
,1-a r .a-1 - ... ,., A-aN , _
Ir^y ; t £x(t)dt| < ^-^f— n-n f or a - o+
hence
lim |J u(x)-u (x) I < n
a-»0 +
Since n is an arbitrary positive number we have proved F.1.4).
To prove F.1.5) we recall the formula (see AB-ST [1970], p. 258,
formula 6.2.2)
! a-1 ,a-i, _ r(a)r(B)
0( X) " r(a+fi) •
We have

98
x t
J6(jau)(X) = r(a)r(BT /((x-tN \ (t-TH-1 U(T)dT)dt
= r(a)r(B) J [U(T) ^ (x-tN_1(t-T)a-1dt]dT
} A-XN-1Xa-1dX
o ^,1/¾ a + 8-1 , _.a + 6 . ,
= r(a)r (fi) J u(t)(x-t) dx = J u(x) ,
by substituting t = t+(x-t)A in the inner integral.
Remark 6.1.2: An analog of Theorem 6.1.1 holds for u£L @,a),
see[Ta,1930], but then the convergence expressed in F.1.4) can only be
stated for almost every x e [0,a]. Furthermore the following continuity
property holds: if u eL @,a) then
a
F.1.6) lim Jau = J ° u in L @,a) .
a->a
o
This can be directly proved by application of convergence theorems for
the integral. Therefore the Abel operator J satisfies the properties
F.1.2 i) , F.1.2 ii), F.1.2 iii) required for being a fractional
operator in L @,a).
For completeness we report the following theorem. We recall that
ex ex
D is the inverse operator of J . Here we have a generalization of this
property .
Theorem 6.1.2: I <j u e L1 @,a) thin
ir 1 -,< 1^0,8 T8-a c „
F.1.7) D J u = J u for B >a.
The fractional operators ja and D have many other operational
properties in common with the operators J and D (where n is a nonnegative
integer). Let us cite,e.g., the fractional integration by parts, see
[Lo-Yo,1938] the fractional derivative of a composite function [Os,1970]
and the fractional derivative of a product of functions (Leibniz rule).
[Os,197Q], [Ro,1975] and [McB,1979] contain many references. It is also
important to know that it is possible to define fractional operators in
spaces of generalized functions, [Ge-Sh , 1964] , [McB,1979], [Ro,1975],
[Ex,1972] and their bibliographies.
6 . 2 . a BxjiaiT Account on Generalizations of Abel Operators
In Chapter 1 and 2 when discussing physical applications we have met
two different types of Abel integral equations. Whereas for determination

99
1/2
of potentials the operator used is J ' according to
F.2.1) (J1/2u)(x) =^/ (x-t)-1/2 u(t)dt ,
\/tt 0
an operator occuring in seismology, stereology, spectroscopic
measurements and in other fields is defined by
R k
//-->-,* f t u(t)dt , _ ,
F.2.2) u -» J '— , k = 0 or k = 1
V^x1
Still other operators of fractional type, operators similar to F.2.1),
F.2.2), are used in other applications. We must resist the temptation
to study in detail every possible generalization, but to a few of them,
interesting for the purpose we have in mind, we should draw the reader's
attention .
Our Abel operators are often given different names in the literature.
For example, the operator J is sometimes called RA.e.mann-LA.ou.VA.lle. (J-tac-
t-ional ope.la.tol o <J olde.1 a, see [Bo,1969], [Li,1832], [McB,1979]. The
operator K defined by
F.2.3) (Kau) (x) =^1-/ (t-x)a~1 u(t)dt
X
is called We.yl ^ia.ctlona.1 ope.ia.toi o (J oldei a, see [Bo,1969], [We,1917].
Among the generalizations of these operators we mention the Erdelyi-
Kober fractional operators and the generalized Erdelyi-Kober fractional
operators, see [Er,1940], [Ko,1940], [McB,1979]. The Erdelyi-Kober
operators are defined by
F.2.4) J^a u = *J^ / (x-t)* t%(t)dt ,
ri oo
F.2.5) K^ u = fj^- / (t-x)a n t n a u(t)dt ,
-n-a oo _
F.2.6) J u = ^y-t-t \ (t-x)a tnu(t)dt ,
J
+
K
, a
,a
u
u
=
T(a)
xn :
r(a) ,
F.2.7) K^a u = T^T J" (x-t)a 1 t n a u(t)dt .
A generalized Erdelyi-Kober operator, see [Er,1965] is defined by
.-n-a . - x _1
F.2.8) J* u = 4 r7IXJ- J" (<J)(x)-$(t))a <j,'(t)<j>n(t)u(t)dt
Operators J , K K , are defined analogously .
n.ra,<J>,n.,a,<J>,n.ra,<J>

100
Evidently the Erdelyi-Kober operators generalize the operators J ,
K of Abel and Weyl type. In fact,
J u=x' J(t'u);K u = x K (t ' u),
and so on.
The operators of Erdelyi-Kober type are interesting for their
relations with integral transforms, in particular with those of Mellin and
Hankel (see [Er,1940], [Ko,1940]). We observe that the operator F.2.2)
for R = +oo and k = 1 is a generalized Erdelyi-Kober operator. In fact,
2 1
posing <j> (x) = x , a = -~, n = 0 in J a ' we identify the operator
F.2.2) as -^ x(J~ ).
The Erdelyi-Kober operators are fractional operators. They satisfy
the relations F.1.2 i) , F.1.2 ii) , F.1.2 iii) with respect to the
exponent a, see [McB,1979]. Many of their properties are similar or
analogous to those of the simpler operators J (see [McB,1979]).
6.3 . Relations between Abel Operators and the Fourier Transform
In this paragraph we study in a formal way the relations between Abel
operators and the Fourier transform. For a general orientation on the properties of
the Fourier transform see the books [Bra,1978], [Ru,1974], [Ru,1985],
[Sn,1972], [Ti,1937], [Ka,1976]. Here we list the most important formulas
for the reader's convenience.
(a) Definition: We denote, the VouKleH tiani&oim o<j a ^anat-ion,
defined on IR , bu
v(C) = J" v(x)e lx^ dx (jet -«><£< + «> .
Remark 6.3.1: There are several other definitions in use, but the
corresponding properties and formulas are trivially related to each other.
1 A
E.g. sometimes v is called the Fourier transform of v, sometimes
A A A A
v(-£) instead of v(£) or even vBn£) instead of v(?).
The most important properties we shall use are
A A A
* w) = v ■ w
wheie v*w meani convolution.
A A A
(b) (v * w) = v ■ w
2 2
(c) The Voun-ien. tia.nA&o>im li, a blject-lon o<j L (IR) onto L (IR) and
II 0 II , = s/THw vll ,
l/(IR) L (IR)

101
(d) The. Vounlnn. -inveu-ion ^oimuta
+00
v(x) = J- J" elx? vE)d5
At first we consider the Abel operator J_ defined by
F.3.1) (J^u) (x) = jr^Iy / (x-t)a-1u(t)dt, 0<x<+».
The corresponding Abel transform of a function u can be written as a
convolution:
F.3.2) (Jau) (x) = (xa*u) (x)
where ( a-1
x c _
for x > 0
F.3.3) Xa(x)
for x < 0 .
ex ex
To study the operator J_ and the related equation J_ u = f the
Fourier transform is very useful because of its algebraizing power (a
convolution is transformed into a multiplication). We obtain
F.3.4) (J^uM?) = (xa*u) E) = (H)-0 uE)
where, for definiteness, we agree upon
(i?) = I SI exp(-ia £ sgn?)•
The Fourier transform of J can illustrate some fundamental proper-
—00
ties of the fractional operators J . In fact, we have
0A A
F.3.5) (J-«>u) <?> = u(?> '
F.3.6) (jl u)(?) = (i?)~1 uE) = F(/ udt) , (here F is Fourier
transform),
F.3.7) (J^J^uJX?) = (i5)-(a + 6)u(?) = (Ja:3'Vu)(?) .
Furthermore the equation J_ u = f can be easily solved by using the
Fourier transform. By J u = f we obtain
-a A A
A5) uE) = fE) ,
hence
A a
F.3.8) uE) = A5)° fE) = (Daf)E)

102
where
D"f(x) = ^ T-f ; HMdt ,
r<1-«) dx -i (x-t)a
As an immediate consequence of formula F.3.8) we get an existence
theorem.
Theorem 6.3.1 : A ne.ce.66a.tiy and iu^-cc-cen-C condtt-Con fion. the exX.6t-
2
ence oft a 6otmt-ion u -in L (IR) oft the equation
F.3.9) Ja u = f
16 that l?|a|f(?)l £L2(IR) an, equlvalently, that
F.3.10) i<2-rr ;fi;a"-axdt<t- .
-oo -oo Ix-tl
F.3.11) II ull , = If I .
l/(lR) a'1
a A 2
For the equivalence between 151 |f(?)l £L (IR) and If I 2 < + °°
see [Ru,1985].
Theorerr. 6.3.1 implies that the problem of solving the Abel equa-
2
tion F.3.9) is ill-posed in L (IR) . To show that this is so , we
2
shall prove that in L (IR) the following statement is valid .
For every f for which F.3.9) has a solution u
there exists a sequence (f ) _ .„, and a sequence (u ) - ._T such that
^ nnsiN ^- n n £ IN
J u =f,f-»f = j u, but u 4 u. In othel wo>id6: In spite of
-oo n n n -oo ' n c
II f -fII , being arbitrarily small for large n, this will not be
IT(IR)
the case for II u -ull „ . Our sequence (u ) ,. will even have
n L (IR) niU
the property II u -ull ~ -» oo for n -»oo .
n ITCIR)
Because J_ is a linear operator it suffices to verify this
statement for f = 0, u = 0. Then for
-1 1
-1/4 a 2 + n -x
f = n ' x e for x > 0, = 0 for x < 0, n £ IN , the equation
J u = f has a solution u £ L (IR) , and both f and u are in L (IR) .
-oo n n n n n
We have
II f II , n = n 1/4 2 1/2 /rBa +h -+0 for n - <*> ,
L (IR )

103
I
n
- a
t
I5I
a +
2
1
1
2
2
d.
1
+ —
n
and via the Fourier transform
1
unE) = n-1/4 T(a +1 + 1) (i5)aA + i5) 2
11 un =i ~ 'V?>|2 d? 2: ^7T <r(a+^ + l)J j
-oo I 5 I >1
- TH (r(a +I + n)J 2"a Vn-oo,
hence II u II -» oo for n -» oo .
n
The practical sign±fjJcance of this situation is as follows. Even if we
2
have in L -norm arbitrarily small error in the data f, the error of the
corresponding perturbed solution may be arbitrarily large (also in the
2
L -norm).
Now we consider the equation with an a priori bound on the first
derivative of the solution. More precisely we study the equation F.3.9)
under the assumption that the solution u satisfies the a priori bound
F.3.12) II u' II 2 < E
L (-00,+00)
where E is a positive number.
2
Theorem 6.3.2: Let f be a g-iven function, 0 £ L (IR) , and let u
and u be the 60Zu.tA.on6 of equation F.3.9) w-ith data ne6pect-ively equal
to f and f + a. I ft u and u 6at-i6&y the a pn.-ion.-i bound F.3.12) we have
a 1
F.3.12') II u-u II , < BE) 1+a II all ,1+a
l/(IR) L (IR)
Proof: By F.3.8) and the Fourier inversion formula we have
1 r .. r. aA , <- - ix 5
u(x)-ua(x) = -± J" (U)aaE)elxs= d?
Therefore, by Parseval's relation,
+00
\ I (u-u ) 'I2 dx = J- \ |?|2A+a) |aE) I2 d5
2n
+00
/ lu-u r dx = J- / I5I IaE)I d?
o 2tt
— CD
The Holder inequality now yields

104
01 1
r i i2 j f 1 r in2 A+a) , A,r, , 2,r\ 1+a I 1 r ,A,r,,2 ,r\ 1+a
J lu-ual dx <!^jj J l?l laE) I d£) I jr J I a (?) I d? J
hence
a 1
lu-u^l 2 < BE)
1+a .. I, 1 +a
a\
2 ,
L (IR) L (-oo.
+00)
Remark 6.3.3: A stability result for equation J_ u = f analogous
to F.3.121) can be found with an a priori bound on u weaker than F.3.12)
In fact, assuming
|ule < El
where
lulQ = [^- \ I5l29luE)l2 d0 1/2
9 V2n m
we have 2^ ^
II u-u II , < BEl)a + 9 II all ,9 + a
l (m) l (m)
The proof is analogous to that of F.3.121).
We now give an approach to the.' Abel equation
F.3.13) (Jau)(x) .=-1./ (x-t)a-1u(t)dt = f(x) for x>0
K ' 0
by aid of the Fourier transform F.3.13) is equivalent to the problem
(X *u)(x) = f(x), x e IR , supp f <= [0,+°
F.3.14) '
supp U (= [0, +«>) .
Here u and f are considered as functions defined on the whole real line,
vanishing on (-=,0).
The difference between F.3.14) and equation F.3.9) is the
condition supp u (= [0,+0=) . It is just this condition that inspires us to use
oneedS: the Paley-Wiener theorems for solving F.3.14) by the Fourier
transform technique. For this theorems see below (and [Ru,1985], [Ta,1973],
[Ka,1976]). We remark that F.3.13) is a simple example of a Wiener-Hopf
equation, see [Wi-Ho,1931] and [Ta,1973] for many references -

105
2
We need the Hardy class H (for general theory see [Du,1970]).
2 2
plane {¢£¢: Im c, < 0} -Ci -in the clan H (IR-) l& theie exlhth a constant
Definition: A function ¢(¢), holomoiphlc -in the lowei complex hal£-
t U £C:
C iuch that
+°° 2
F.3.15) \ |$(£ + in) I d£ < C ^on eveiy n < 0.
— oo
2 2 2
I<J $£H (IRJ we dt{,int ai tuace o<J $ *fie junction $_£ L (IR) inch
(i) ¢.E) = lim ¢E + in) almost eveiywheie.
n-»o~
+°° 2
(ii) .lim_ / I $E+ in) - $-E) I d? = 0 .
n->o -c°
Without proof we state the well-known Paley-Wiener theorem.
2 2
Theorem 6.3.3: (I) Suppose that $£H (IR- ) and define a faunct-ton
2 A
f £L (-00,+00) ai, Invelie Tounlen. tuanb^oim o<j fE) = ¢-E) , wheie $—04
tfie tKa.ce. o<5 $ on .the -tea£ axxli. Then supp f<= [0,+00),
2
(II) Ifj f £L (-00,+00) and supp f<= [0,+00) then theie ex-ihth a unique
function $£H (IR_) iuch that $_ = f .
As an immediate application of this theorem we have
Theorem 6.3.4: Equation F.3.14) hat, a unique solution In
L2(IR) li and only l{> (ic)af(i;! £H2(IR2).
Take the Sobolev space hS(-oo,+oo), s £ IR as the space of temperate
distributions v (see [Ge-Sh,1964]) such that
■) + °° ? <5 * ?
II vll ^ =/A+ 5 ) lvE) I d5 <
H (-00,+00) -00
(k) s 1
Then for example 6 £H (-00,+00) for s < —j - k .
0
(k) k -lxo
In fact, from 6 = (i5) e we conclude
0
Il6xk) II I = J" A +52)s 52k d5 <+»
0 H (-00,+00) -00
for s <- ^ - k .
Theorem 6.3.5: Let f £ HS (--00,+00) , s £ IR , supp f <z [0,+00) .
Then the unique solution u o<J F.3.14) In the ipace o<J tempeiate dlhtn.1-
butloni li In the Sobolev ipace H (-00,+00) and

106
F.3.17) II u|| < || f ||
H IK) HS(IR)
To prove this theorem we use a generalization of the Paley-Wiener
theorem (see [Ta,1973; Th. 1.3.3, Le. 1.3.4]), namely
Lemma 6.3.1: Let f SHs(-»,+»|, seJR.Tke.yi the two pKppobltlont, [I]
and {-i-i} ale equivalent.
(i) supp f <= [0, +«>) .
(ii) Theie exliti a unique function ¢(£) with x.he following ^aun. piopei-
t-iei
(a) ?-»A +ic)s ¢(¢) -ii « H2(m?) ,
A
(b.p f(?) Ii, the tiace o<j ¢(¢) ,
+°° ? d a ->
(b2) \ A + 5 ) If (?) I d£ < + » ,
(b3) lim_ J" A + 5 ) 10E + in)-f E) r d5 = o .
r]->o -oo
Remark: By IR- we denote the lower half-plane {?l Im ? <0}, and
generally ? = 5 + in with 5 £ IR , n € IR .
proof of theorem 6.3.5: By Fourier transformation we see that the
equation Ja u = f is equivalent to (i£)" a uE) = tE). Thus we obtain
A n A c
uniqueness and uE) = (i?) fE). Now f £H (-00,+00) and supp f a [0,00) .
Hence f(£) satisfies (ii). Furthermore we observe that (a) is equivalent
to the statement "The function <j>E) of which f(£) is the trace is holo-
morphic in the lower half-plane {Im 5 <0} and
+r°° 2 2 s A 2
F.3.18) / A+?+n)lf(S)ld?<C for n <0."
—00
In F.3.18) we have written f(?) instead of ¢E), and we shall,
for simplicity , do so on other occasions. Remember, furthermore,
C = 5 + in .
s-a A 2 2
We see that the function C-» A +i?) u(C) is in H (IR_) . In fact,
A A
u(?) can be holomorphically extended as u(C) to the lower half-plane. For
every ri < 0 we then have
F.3.19) \ (i+s2+n2)s-alu(?)|2d5 < \ d+c2+n2)If (?) I2 d5 < c .

107
Hence (a) of Lemma 6.3.1 is satisfied if we pose s-a instead of s, u
instead of f and <j>.
The properties (b..) and (b.,) are immediately seen as valid. To
show (b_) we prove
+00
F.3.20) lim J" A+5K-° I (i?)a f E)-A5)° fE) I2 d? = 0
r)->0- -00
as follows. First we estimate
T d+52)s"ai(i?)a r(?)-(i?)a f(ai2 aeN V2
/ +00
<
(/ d+52+n2)slf (c)-f E) I2 de) 1/2
/ A+5 )S~al (i?)a -(i5)a| lfE)!2 d?> 1/2
Then for n£ (-1,0) we observe that
+r°° 22sA A 2 T°° 2sA A 2
\ d+r+n ) if <?)-f E) r < 2 / (i+rrif(c)-fE)i d? .
—00 —00
From this (since f EH (-00,+00) and supp f c [0,+00) imply (b..) with
$(C) = f(?) ) follows
T 2 2 s A A 2
lim / A+5 +n ) If(?)-fE)I d? = 0 .
n->0_-oo
Finally, by applying the dominated convergence theorem to the integral
+00
J" A+?2)S-a |(i?)a_ (i?)a||f(?)|2 d? , we obtain F.3.20).
~oo
Conclusion: u satisfies (ii) of Lemma 6.3.1, therefore u£Hs-a(IR) ,
supp u c [0,+00) and, see F.3.19),
Hull s_a < II fH s
H (IR) H (m)
6.4. Relations between the Abel Operator and the Mellin Transform
To study the relations between the Abel operator and the Mellin
transform we use the Abel transform as modified by Erdelyi-Kober (see
§ 6.2) -a x ,
F.4.1) (J+0@u) (x) = fjS) / (x-t)a u(t)dt, x>o .

108
This relation can be also written in "multiplicative convolution" form;
+00
F.4.2)
0
where
(J0,au)(x) - I Ga(!)u(t)£F =: Ga8 u
F.4.3) GaE) = '
r(aMaE-D1 a
for ? > 1
for ? < 1 .
The Mellin transform ( M v) (s) of a function v, defined on [0,+=°)
is given by
s-1
F.4.4)
( M v) (s) = J v(x)x& dx , s 6C ,
0
see [Bra,1978], [Ti,1937], [Sn,1972], wherever this integral converges.
For abbreviation we often write v*(s) instead of ( Mv)(s).
For the reader's convenience we list the most important formulas of
Mellin theory. For operating with the Mellin transform we need a strip
a. < Re s < a? viiere ( M v) (s) is a holomorphic function of s. If, e.g., v
is continuous in [0,=) and v(x) = oA/x) for x->» we can take the strip
0 <Re s<1. This condition can, of course, be considerably relaxed.
In the sequel we shall at several places tacitly use the Mellin
inversion formula (see [Sn,1972], [Ti,1937].)
c + i«> _ . +0° _ , ..,
(a) v(x) = --U f v*(s)x s ds = 4~ \ v*(c+it)x lc+lt'dt
C—loo —00
where the real number cE @.,0.,). We shall take c = 1/2
shall use
(b) M(/ u(£)v(tJi;s) = (Mu)(s) (Mv)(s) ,
0 ^
Likewise we
(c) M( xv'(x)
-s( Mv)(s)
? 1 +°° 1 ?
(d) / I v (x) T dx = y- / I v* (y + it) r dt .
2n
We may call property (d) the Parseval equality.
We obtain
(J* u)*(s) = G*(s)u*(s) .
u , ex ex
Now by using the beta function we have
_ rA-s)
GS(S>
r (a + 1-s)

109
therefore
F.4.5)
(J0,au>
(s) =
rd-s)
r (a+1-s)
u*(s)
Using this formula and the properties of the r-function we find suffi-
2
cient and necessary conditions for solvability in L @,+=°) of the
equation
F.4.6) J^a u = f .
Theorem 6.4.1 ([Ko,1940]) :A neczAiaAy and Au£&i.cU.ent condition fcoK the
2 2
equation F.4.6) to have, a iolwtlon u£L @,+«) am that w£L (JO, uhexe
F.4.7)
w(t) = A+|t|)a( Mf) (± + it)
Now
Proof: Let u£L @,+=°) solve F.4.6). By F.4.5) we have
u*(s) = Ii«llls-)f*(s)
U lS' TA-s) r ls' •
+°° ? +°° ,
J u^(x)dx = 2¼ J luMj+it)! dt ,
hence by F.4.5
F.4.8)
2 1
\ u (x)dx = -=^ \
0
2n
r(a+^- it) 2
r(^ - it)
I ( Mf) (i + it) |2 dt
To proceed further in our proof we need a lemma.
Lemma 6.4.1: Thete ex-Lit two poiAlt-Lve. conhtanth c. = c. (a) and c~ = c?(a)
iixah that
F.4.9)
c1 A + |t|)u <
r(a +j - it)
r(^- it)
< c2d + iti;
Proof of the lemma: We recall that for every positive number a and real
number t
n i j- i 1
(i) lim | r(a+it)Ie |t|
|t|-w>
\/2ti
(see [Ab-St,1970; p. 257, for. 6.1 . 45]),
(ii) |T(a+it)I < T(a)
(see [Ab-St,1970, p. 256, for.6.1.26]).

110
Furthermore we have:
2 2
(iii) I r (a + it) I > I r (a) I exp {- -^- (-L +-IL.) } .
a
To see that (iii) is valid vve use formula 6.1.25 p.256 of [Ab-St, 1970]
2
I T (a + it) I '
IT(a) I'
n L
n=0 i+JL
(a+n)'
and we obtain
log
I log A+-
IT(a + it) I _
I :: (a) \2 n=0 (a+n)'
y t2 . +2,1 y 1 ,
n=o (a+n) a n=1 n
J. . 1 TT
-t (— +T-
a
hence (iii).
The inequality F.4.9) is a consequence of (i), (ii) , (iii)
We now continue the proof of the Theorem 6.4.1 .
By F.4.9) and F.4.8),we get
/ |w(ti T dt < =j- \ u^(x)dx < +oc ,
-oo C 0
2
hence w £ L (IR) .
2
Conversely let w£L (IR) where w is given by F.4.7). Take
F.4.10)
. +°° r (a + ^ - it)
U(x) = i ; 2
r(^-it)
f* (-j + it)x
— -~-it
dt
and from
and
r (a +-1-it)
u* A + it) = — - f * {± + it)
r(^ - it)
/ u^(xldx = ± /
0
2n
r (a +-1- it) 2
f-i f*(l + it
r(^-it)
dt
2n
/ |w(t) T dt <
conclude that u£L @,+oo)

111
From F.4.10) and F.4.5) we deduce
(J0,a u-f)*(-J+it) - 0 .
Now u e L @,+=) and f £L @,+=) imply J* u-f € L @,+=) and the
inverts , ex
sion theorem J u = f, i.e. F.4.6). The proof is completed.
The Mellin transform can be used for obtaining a stability estimate
for the equation F.4.6) under the assumption of an a priori bound
F.4.11) || xu" II 2 <E and lim u (x) = 0
L @,+=) x->+=
where E' is a positive number.
Theorem 6.4.2; I
iatli^lei the e&t-lmate
2
Theorem 6.4.2; I j$ f £ L @,+=) the iolutlon u o j$ the equation F.4.6)
a 1
F.4.12) Hull 2 < C(a) llxu'll 1+a ||f||l+a
L @,+=) L @,+=) L @,+=)
wheie C(a) It, a constant depending only on a.
The estimate F.4.12) gives a measure of the stability of the equation
F.4.6) when the solution u satisfies the a priori bound F.4.11). In
fact,if u.,u2 are the solutions of F.4.6) corresponding to data f. and
f2 and satisfying the a priori bound F.4.11)
we have
a 1
llu1-u2ll < C(a)BEI+a II f1-f2ll 1+a
L2@,+=) L2@,+=)
Furthermore we have the following stability estimate for Abel's
equation
F.4.13) Jau = f .
Corollary 6.4.1; If, the solution u oj$ F.4.13) bat-Lbi-ieb the a pi-ioi-L
bound
II xu' II < E
1/@,+=)
we have
a 1
+=
F.4.14) II u|| - < C(a)E1+a(/ Ix a f(x) |2dxJA+a)
L @,+=) 0

112
Proof of Theorem 6.4.2: Since
F.4.15)
(J* u)*(s) = FA S) u*(s)
U,a r(a + 1-s)
and F.4.6) imply
F.4.16)
u*(s) = Ha + l-s) f*(s)
rd-s)
we have
■) 1 +
F.4.17) J u (x)dx = y- J
2n
r (a +-J - it)
T(\- it)
f*(-l + it) I2 dt .
Furthermore
(xu') *(s) = -su*(s) ,
hence
F.4.18) J (xu'J dx = 2^ J (t+j]
o -°°
T (a+-~ - it)
T(\- it)
If* (-^ + it) I2 dt .
Now by Lemma 6.4.1 we have
F.4.19)
c1 (a) A + Itl)" <
r(a +-J - it)
r(l- it)
By F.4.17) and Holder's inequality we obtain
< c2(a)A + Itl
J u (x)dx < ^2n J (.1 + Itl) ^ I (M f) (^ + it) r dt
0 —oo
+ 00
< c2(aMlL ; A + |t|J(a + 1)lMf4 + it)i2 dt)a + 1.H
with
h = (J- ; i <m f) <i + it) i2 dt
1 +a
l2n
hence by the left-hand inequality of F.4.19)
A + itiJ(a+1' < -j
r (a +^-- it) .2 9 1
-r-^r:—I (t +t'
c^(a) ' T(^- it)
and, recalling F.4.18),
j u2(x)dx < C2(a) (J (xu'JdxI+a (J f2dxI+a
0 0 0

113
a
where C2(a) =0^ (a) (-^ ) 1+a
e1 (a)
6•5. Some Relations between Abel Operators and Hankel Transforms
We now study the relations between Abel tnayn faonmi oj$ one oj$ the fallowing typu:
x
F.5.1 i)
Au(x, =_L j tu(t)dt ^ x>0 f
F.5.1 ii) A2u(x) = 4: 7 fc U(t)dt , x>0 ,
^n x /t2-x2
and the Hankel tnanifaonm oj$ olden, zeno. The function u is here assumed
as continuous. For more general definitions and properties,we refer the
reader to [Sn,1972], [Ti,1937], [Bra,1978].
The Hankel transform of order zero, H , is defined by
F.5.2) HQu(x) = J 5 u!£)J0Ex) d?, x>0 ,
where J denotes the Bessel function of first kind and order zero
o
(see [Co-Hi,1953, Ch. VII] and [Wa,1944]). We begin by recalling the
following formulas (see [Sn,19 72; p. 518])
F.5.3 i) J aOS ^ dt = 1 ti J (x?) , x >0, ? >0 ,
0 JF~^
t
+oo
F.5.3 ii) J Sln gt dt = -1 t J (x5), x > 0, ? > 0 ,
X t J (t5) .,
F.5.3 iii) / — dt=-sinx£,x>0, ? > 0 ,
0 v^U1
=° t J(t?)
F.5.3 iv) J — dt = - cos xt , x > 0, ? > 0 .
We also need the definition of the VounZen h-ine and coilne tn.aniion.mi.
+°° A
F.5.4 i) F (u) = / u(t)sint£ dt = -2i u_(£) ,
0
T A
F.5.4 ii) F (u) = / u(t)cost£ dt = 2 u (?)
C 0
where u is the odd extension of u:

114
u (x) = <
u (x) x > 0
-u(-x) x < 0
and u is the even extension of u
rU (x) X > 0
u+(x) =
Lu(-x) x < 0 .
The operators F , F , A,, A~ and H are connected as follows,
r s c 1 2 o
F.5.5 i) F^ = f Ho ,
F.5.5 ii) FcA2 = f Ho .
To see that F.5.5 i) holds observe that by F.5.3 ii)
+=° , t
(F A.u) (x) = J A.u(t)sin t x dt = / — J gu(g)dg sin tx dt
0 ' 0 Vn 0 /t2_r2
L J {?u(?) J sintxdt^ dc = Vj- ; ?u(?)Jo(x?)d?
^n 0 ? v^ " °
= ^r (HQu) (x) , x >0 .
Analogously F.5.5 ii) can be shown to be true.
Many other relations between Hankel, Fourier and Abel transform can
be found by direct use of formulas F,5.5 i), F.5,5 ii)and the properties
of Hankel, Fourier and Abel transforms, we do not present a comprehensive
list of all possible relations.
The interested reader is advised to find many other relations by looking
into large integral tables, e.g.,-, those of [Gr-Ry, 1980 ] , [Ab-St ,1972] .
From the well-known relations (see[Sn,19 72]) and the four formulas
F.5.3)
Fs = 1 Id ' Fc - 1 Id ' Ho - Id
we find:

115
F.5.6 i) Fg HQ = \/n A1
F.5.6 ii) F HQ = |/n A2
F.5.6 iii) HQ Fg A = ^- Id,
F.5.6 iv) H F A0 = ^- Id .
o c 2 2
As a simple exercise the reader should prove F.5.6 i) and F.5.6ii) by
using F.5.3 iii) and F.5.3 iv) .
6.6. Some Relations between the Plane Radon Transform and the Abel
Transform
In this paragraph we briefly discuss a few formal relations between
the plane Radon transform and the Abel transform. We explain some ideas
of two papers of Cormack A963,1964) (see also the paper of Cormack in
[I.E.E.E. 1982, pp. 35-42]).
First we recall the definition of the Radon transform of a function
of two real variables. We refer the interested reader
to the literature, in particular,to [He,1980] and [Lu,1966] for a
theoretical treatment of the subject and [I.E.E.E. Proc. 1982], [Lo-Na,1983],
[Na,1986] for the mathematical and numerical problems related to the
inversion of the Radon transform.
2
Let f be a function defined in the plane IR . Take L „ as the
^ p,0
2
straight line {(x,y) £ IR : x cos 0+y sin 0=p} where 0£[O,2n), p is
a positive number (see Fig. 6.6.1) .
The Radon tiani&olm of f is defined by
F.6.1) (R f)(P/Q) = J f(x,y)ds
L0,p
where ds is the element of length.
Let the support of f be contained in the unit circle
D = {(x,y): x2 + y2 < 1 }. If we interpret (p,9) as polar coordinates in a
plane, then it is obvious that also the support of R£ is contained in the
unit circle.

116
> x
Fig. 6.6.1
Writing f(r,ip) instead of f(r cos ic, r sin ip) we have the Fourier
expansion
F.6.2) f(r,u>) X f (r)eimp , 0 < r < 1, 0 < id < 2tt ,
n=-°°
with
F.6.3)
fn(r) = jy / f (r,<p)e imf> dtp .
Observe that the equality F.6.2) holds in the sense of uniform
convergence if f is continuous and
F.6.4)
id f I
u 3tP '
This can be proved by a simple modification of the proof of uniform
convergence of Fourier series (see [Ka, 1976; Ch.I,Tl]).
Now, to calculate the integral / f(x,y) ds in the polar coordinate
Lp,0
system (r,tp), use (see fig. 6.6.2)

11
7
> X
Fig. 6.6.2

118
= v^1
(i) s = \/r -p
r dr
(ii) ds
ri 2
Vr -p
(iii) ^ = cos{0-tp)
and conclude
J f(x,y)ds = J f(r,tp) r dr + J f(r,tp) r dr
LP0 MP1 JrW P2M J*W
By the Fourier expansion of f and the relation (iii) we have
+=° 1 , • p.
F.6.5) J f ds = 2 X J f (r)cosln(ip-0)) Q e nu
L „ -oo p /~2 2
p0 r Vr -p
Now the definition of Chebyshev polynomials of first kind ([Gr-Ry,1980;
p.1032]) implies that the integrals at the: right hand side are equal to
1 T (£)r dr
F.6.6) J f (
n r
P ir
2 2
With g(p,0) = Rf(p,0) we obtain by F.6.5) and F.6.6) the equations
1 t <R\ r
F.6.7) g (p) = 2 f f (r) nlr' r , .^
yn r J n dr , n££ ,
p n-^i
vr -p
for g (p) = J g(p,0)e in°d0 .
0
We observe that F.6.7) is an Abel equation different from the usual
form studied in this book. But also in this case it is possible to find
an explicit expression for the solution. In fact, if we multiply equation
F.6.7) by T (p/z)(z/p) and integrate from p = z, to p = 1,
HZ 2
vp -z
interchanging the order of integration on the right hand side, we obtain
1 z T (p/z)g (p) 1 r r z T (p/z)T (p/r)
F.6.8) J n n dp = 2 J f (r)dr J —5— —2— dp .
z /1 2 z n z /1 2 /~2 2
p vp -z p vr -p VP -z

119
Now (see Lemma 6.6.1 below)
r Tn(p/z)Tn(p/r )
2 r z / ———-————- dp = n ,
z /~2 2 ri 2~
pyx -p vp -z
therefore
1 r g (p)T (p/r)
F.6.9) f (r) =-1 #- f 2 2 dp .
n ti dr J /-^ =- ^
r / 2 2
P VP -r p
Lemma 6.6.1: For nonnegative integers n we have
r T (p/z)T (p/r)
F.6.10) rz \ —n dp = ^ .
z /~2 2 ^1 2
pyr -p vp -z
Proof: In the integral which we denote by I we substitute
cos <j> = p/r and note 0<<)><tt/2. Because p/z > 1 there exists a value 0>O
with cosh 0 = p/z. we have cosh 0 = (r/z) cos <j) . In the calculation of
I we write, to simplify the notations, 0,()),p as independent variables.
Recalling T (cos x) = cos nx and cos (ix) = cosh x we see, by analytic
continuation, that T (cosh 0) = cosh n© .
Therefore
and
F.6.11)
I
n
dp_
P
C cosh n0 cos n<j)
z sinh n0 sin n<j)
_sin <j) ,, _ sinh 0
cos <j) * cosh 0
.d£
P
d0
Now
r
t t f cosh (n + 1H-cos(n + 1) ()) -cosh (n-1) 0-cos (n-1) $ dp
z sinh 0 • sin())
The trigonometric and hyperbolic addition formulas yield
cosh(n + 1H-cos(n + 1) § - cosh(n-1H-cos(n-1) §
= 2 (-cosh n0 •• cosh0 • sin n<j) sin <$> + sinh n0 sinh 0 • cos n ()) cos ()) )
Therefore
 A i-1 _i) = / /'sinh n 0 cos n <p cos $_ cosh n 0 cosh 0 sin n (f\ dp
n+l n ' z V sTn~$ sinh 0 ^ p

120
Now
sinh n0 cos ni> cos
cosh n0 cosh 0 sin n $ \ dp
sinh 0 / P
(sinh n0 cos n<j) d0 + cosh n0 sin n<J) d<J)
d (sinh n0 sin nd)) .
Finally
•2<In+1 - In_1) = sinh n0 sin nd
p = z, 0 = 0
p = r,
That is
F.6.12)
1,,=1 , for every n >
n+1 n-1 J

121
Now we calculate I and I. . With the help of formulas 3.198 and 3.191-3
o 1 r
of [Gr-Ry,1980] we obtain
I
r
c zr dp
J
z / 2 2 / 2 2
p yr -p Vp -z
- -j J
0
rz dt tt
O _ ,-^--^ r^-^ , 2 2 2
[z +t(r -z ) ]\/1-t \/t
r
f
z
P dp
/ 2 2/2 2
\/r -p VP -z
1 ] dt
2 0 \/t VT=T
tt
2
Hence
*1
I = -=r for every n
n 2 -1
We observe by F.6.7) that the inversion of the Radon transform is
an ill-posed problem. In fact if f and g are radial symmetric functions,
then the inversion of the Radon transform is reduced to an Abel equation
g(p) - 2 } JdlLL. dr
and, as we shall treat in detail in Chapter 8, the Abel operator,, when
acting from L to L , does not possess a continuous inverse.
Here we do not study the problem of stably inverting the Radon
transform. We only observe that by Fourier expansion F.6.2) and by the
infinite sequence of Abel equations F.6.7) it is "in principle" possible
to find some stability results for the equation Rf = g, assuming an
appropriate a priori bound on one of the derivatives of f. These results could
be obtained by arguments similar to those used in Chapter 8. However, this
method doesn't seem very convenient (see also [Lo-Na,1983; VII-C]) and
more general techniques can give optimal results (see [Lo-Na,1983;th.6.2]).
Appendix 6-A. Generalized Abel Equations: Survey of Literature
We briefly consider here a few aspects of generalized Abel equations
introduced in Appendix 3.A. In essence we give a brief survey of published
results with comments.To the readers interested in studying in depth the
arguments we recommend [Ca,1922], [Ga,1966], [Me,1978], [Sa,1960] ,
[Wa,1979] the papers of Peters A969), Samko (several publications),
the book of Meister A983) and the dissertation of Penzel A986).
Remember the form (see C.A.1))
F.A.1) Mu(x) = ♦(x)(j"u)(x) + i|i(x)(k"u)(x) = f(x), a<x<b , of the
ci JD —

122
0(, Ot 01
generalized Abel equation. Where K, = (J ) (the adjoint of J with
b b a a
respect to the scalar product (u,v) = J uvdx) and a £ @,1) .
a
The function <)>,i|j and f are known, the function u is unknown.
Furthermore l<J>(x) I + |t|j(x) I > 0 for all x £ [a,b].
Sakalyuk in [Sa, 1960] assumed that the functions <)> and i)j , in
F.A.1) are H61der-continuous and that with a positive e and a function
f* that is ,together with its derivative ?', Hc-lder-contimious, the function
f has the form
F.A.2) f(x) = [ (x - a) (b - x)]£ f(x), a < x < b.
He looked for solutions of the form
F.A.3) u(x) = ^^
[(x-a) (b-x)] a a
where a > 0 and u is Holder-continuous. If u is of this form,the
function
F.A.4) *(z) = [(z -a) (b-z)]"a/2 J u(t>dt z£C,
a . . , 1-a
(t-z)
with the powers suitably defined, is analytic in C--[a,b], and the
functions $(x + i0) = lim $ (x + iy) , are locally H61der-continuous on [a,b],
y->0+
Now the integrals
F.A.5) ^ u(t)dt b u(t)dt
a (x-t) x (t-x)
can be written interims of 4>(x+i0) and $(x-iO). Hence it is possible
to reduce F.A.1) to the equivalent equation
F.A.6) $(x + iO) = A(x) $ (x - iO) + B(x) ,
where A and B are known functions depending on $ ,i> and f.
F.A.6) is a boundary value problem of Hilbert-Riemann type . ([Ga,1966]) .
Other authors, for example Samko A967 - 1969) and Peters A968,
1969), investigated the equations F.A.1) in a different way, namely by
using integral formulas of the type (see [Sa,1, 1967])
F. A. 7) Kab u = cos(aTT) Jau + i sin (cm) (b - x)a Sab[ (b - xfa Jau] ,
where (the integral being taken as Cauchy principal value)
F.A.8) S .u(x) =4- J V(t)?f •
abv/ TTi'x-t)
a
Using the transformation F.A.7)they reduce the equation F.A.1) to a

123
singular Cauchy-type integral equation of second kind, in the unknown
J u .
In a similar way it is possible to study the equations:
b
F.A.9) Mu(x) + J T(x,t)u(t)dt = f(x) , (a < x < b),
a
b
M*u(x) + J T(x,t)u(t)dt = f(x) , (a < x < b),
a
where M is defined by F.A.1), M* is the adjoint of M, and T has a
singularity of order strictly less than 1 -a (see [Sa.2, 1967], [Sa, 1968]).
In [Sa, 1969] Samko studies the equation F.A.1) with a = -co, b = +«>.
He further finds some interesting relations between the equation F.A.1)
with a = -oo, b =+o= and other integral equations of convolution type.
Concerning systems of generalized Abel equations, we quote the works
of Lowengrub-Walton, 1979 and Walton, 1979 • they consider systemsof the
form:
F.A.10) fr (X) 1U(t6)dt6 + ^(x) J ^t = f (x), (a<x<b),
a (x - t ) x (x - t )
F.A.10i) 4, (x) J Hiil^t + „, (x) h{l)dts = f2(x), (a < x < b),
a (x - t ) x (x - t )
where 0 < a < 1, 5 > 1 , <j) , <j> , tjj , tjj , f , f., are known functions with
properties similar to those of the known function of equations F.A.1),
u and v are the unknown functions.
Appendix 6.B. A Modified Abel Transform
In this appendix we study, in more detail , the modified Abe.1 tK&nb-
^o Km
F.B.1) I1/2u(x) := (J+n 1 u) (x) = —J U(t)dt 0 < x < +°° ,
U'2 /ir o /x(x - t)
and the related equation
F.B.2) I1/2u(x) = f(x).
By formulas F.4.1) and F.4.2) we see that
F.B.3) I1/2u = G1/2 « u
where © is the multiplicative convolution and
1
F.B.4) Gi/2(?) =^ VttU?-1)
for 5> 1
0 for 5 < 1

124
We have, see F.4.5),
F.B.5)
(M I1/2u)(s)
TA-s)
r(f-s)
Mu(s)
1/2 2 2
Theorem 6.B. 1 : I ' : L @,+«>) -»L @,+«>) i.4 cont-Lnuoui and
F.B.6) II I1/2H2 = V'n
2
Proof: For any function u £ L (IR) , we have
F.B.7) II I1/2ull^ = 4- i tghjLLl ( Mu) (i+ it) I2 dt .
In fact, by Parseval's equality (see(d) ,Chap.6.4) and F.B.5), we have,
with || || = || ||
IT(IR)
1/2, ,,2 _ 1
II I " ull
2tt
I 1/2 1 I 2
J ( M I 'u) D- + it) dt
+- r (,-it) ,2
4- f — M u D + it) dt .
2n
r(i - it)
Now (see [Ab-St,1970; formulas 6.1.23, 6.1.29, 6.1.31])
H-j- it)
r(i - it)
T(-l - it)T(-l + it)
r d-it)r(i+it)
tgh n t
t
Hence F .B.7) . From
d tcjh_z_ _ _ 2( h z) 2 J sinh t
dz z • o
we deduce that tgh z/z is increasing in (-=,0) and decreasing in @,+
hence assumes its maximum 1 at z = 0. By (C.B.7) and property (d) of the
Mellin transform we have
hence
m -rV2 I, 2 1 .tgh tt t, (
II I u 11 2 < j max ( * t—) J
IR n -<*
II I1/2ull2 < ^tTII ull2 .
(M u) (-J+ it)
dt = nil
In this estimate v'tt cannot be replaced by a smaller constant. To
this consider the sequence (u ) defined by

125
,,-1/2
un(x) = -JUmi sin A in x) , n e M ,
v'x In x
and calculate
V'2 - 1 '
II I1/2u ||2 = n ^11 t2hn_t_ dt ^ „ .
^ 0
2
Lemma 6 .B. 1 : I <j lim u(x) = 0 and xu' £L @,+=) we fiave
x->+=°
F.B.8) llxu'll2, =4-/ (t2+i) | ( Mu) A + it) I2 dt
l/@, + =°) ZTT -co ' z '
2 —
Proof: Since xu' EL @,+=°) we have lim v'x u(x) = 0 ; furthermore
x->0
from u (x) -» 0 as x -»<=/ I u' (t) I = — |tu'(t)l and the Schwar z inequality
we have .
_ +=° +=° ., 9 7
V'xlu(x) I < v'x / |u'(t)|dt < (/ t u' (t)dt) ->0 for x -> + =° .
Hence lim Vxlu(x)I = 0 and
x->+=°
F.B.9) M(xu')(s) = -s Mu(s) if Re s = j .
F.B.8) follows from Parseval's equality and F.B.9).
2
Lemma 6.B.2: I & xu' £ L @,+=°) and lim u(x) = 0 then
x->+=°
F.B.10) || uli < 2 II xu' II 2
L @,+=°) L @,+=°)
tht constant 2 -In the -Inequality F.B.10) be-Lng beit-poa-Lble.
Proof: Integrating by parts, using v'x u(x) ~> 0 for x -»0
and for x->=°, and applying the Schwarz inequality, we obtain
1 1
+=° ., +=° +=° ., -j +=° -, -x
F.B.11) / u (x)dx =-2/ xu'udx < 2(/ (xu') dx) (/ u^dx)
0 0 0 0
hence F.B.10). In order to see that here 2 is the best-possible constant
let

126
u (x)
n
1
v'nx In x
sin (— In x ) for n£ B,
Each function u satisfies the hypotheses of Lemma 6.B.1, furthermore
(M un) (-2 + it) =«
1 for |t| <
0 for |t| > -
v - n
Hence
F.B.12)
+°° o +oo _ 2 +oo ?
J u^dx = J- J I ( M u ) (-^ + it) I ^dt = -^2— J (xu; ) ^dx
0 -oo 3n +4 0
F.B.11) and F.B.12) imply
sup
- 2 1
(J u dx)z
J2
+~ 2 1
(/ (xu'rdxr
2 .
Theorem 6.B.2; Let u be -Cfce 4o£u-C-coia of the equation F.B.2) and let
£ be a function iuch that u(x) -» 0 as x-»°°. Then the e&tlmate
F.B.13)
llull2 < (l+AI/6 || fii^3 ||xu'll21/3
holdi,.
Proof: The function
F.B.14)
(X) = X(l--jJ tgh(n(l-lJ) , 0< X <4,
is increasing and convex on @,4). By Jensen's inequality [Ru,1974]
J_ J I ( Mu) (^-+ it) r dt
2n -«>
we obtain
II xu' II2
1 ? 1 I 1 I ^
2^ / (t +^) | ( Mu) (^+ it) | dt
iW $( 2 1} (t +V ( Mu) (^ + it) p dt
-oo t +
2
Hu II?
H£ll2
2n
; <t2+l) (Mu)(-i + it>
dt
llxu' Il2 llxu' ll2

127
Therefore
F.B.15)
Hull 2 < llxu' III * 1
f\\ fii:
Vll xu' II2
( 1 1
Putting y = tt i y - -j\2 , we get the following chain of inequalities.
i(X) =iAtghu = TTAf1 _2
e"K + 1
ttX
2ttX
3/2
U + 1
ir/T^X + 2/ X
Now, the identity max (/4 - X tt + 2/X) = 2 /4 + tt implies
[0,4]
1
()) (X) > TT (TT2 + 4J X3/2
therefore
F.B.16)
for 0 < X < 4,
1/3
a-1 , i f 4 ... 1V'"" 2/3
* (U - V —2 '
From F.B.15), F.B.16) we conclude
llull2<
TT
+ 1
1/«
llfll2/3 11 xia' 11 2 / 3 ■
Remark 6.B.1 : The estimate F.B.15) cannot be improved. To see
this we show that
F.B.17)
sup | Null2, : 11 xu" II2 = 1, II I1/2 ull2 < e } = <|> 1 (e)
if 0 < e < 4tt (we put this bound one since, by Lemma 6.B.1
2 2
Hull 2 < 4tt llxu' II 2 , and 4¾ = f D) = max{()) (X ) |0 < X < 4}) .
If llxu'll2 = 1 and II I1 /2 u II 2 < e then
F.B.18)
II u II2 < <)> 1 (e)
Furthermore, let
where X =
F.B.19 i)
un(x)
(e) = (
y2n ,n+Xy 1
o ■ 2 + , 3
2tti n 3n
1
2
J
1, .
-s
x ds
Y+iy
2
1 1.2
We get
II unll2 - $ 1 (e) ,
F.B.19 ii) II xu' II, = 1,
n <i
F.B.19 iii)
II Iu II2 < e .
n 2. -

128
To see F.B.19 iii) consider
II lV2u II? = f 1— ) n /+n SLEt dt
nA + ,„2
< — tgh ny = $ (A) = t
The equality F.B.17) follows by F.B.19 i), F.B.19 ii) and F.B.19 iii)

Chapter 7: Nonlinear Abel Integral
Equations of Second Kind
7.1. Introductory Remarks
A survey will be given of the evolution during the last 35 years of the
analysis, applications and numerical methods in the field of izcond
k-ind nonlZne.a.1 Abe.l-type. -lnte.gtia.1 tquatloni. Nonlinear Abel integral
equations of first kind have been treated in 5.2 with regard to
existence and uniqueness of solutions. The main analytical tool for second
kind Abel equations is, quite naturally, the Picard iteration procedure
(or a corresponding fixed point principle, either of Banach or of Edelstein
type. .To develop the basic ideas in the simplest setting we first
treat, in 7.2, Unw/i Abel equations of second kind.
In the literature one can roughly discern two directions of research on
nonlinear Abel integral equations of second kind:
(i) analy&^&-motivated LnvehtLgatLonh,
(ii) applLcatLoni,-motivated Znvlitigation 6.
For (i) see [Di,1958], [Re-St,1971], for (ii) see the pioneering papers
of [Ma-Wp,1951], [Ro-Ma,1951], [Pa,1958], [Le,1960], [Ol-Ha,1976].
The applications-motivated investigations, however, are also highly
analytically minded in the way they are carried out.
They are mainly motivated by bounda/iy dlHuilon-radiation psioblemi.
And (iii) there are many papers on Au.men.lcal me.th.odi, some of which are
inspired by the works of Mann, Wolf and Roberts mentioned
earlier.See [Gr,1982], [Gr,1985] and also [Ke,1982], [Li,1967], [Li,1985],
[Mi,1971], [Mi-Fe, 1971 ] , [Ri,1982], [Ha-Lu-Sc,1986], [Ha-Lu, 1980] ,
several papers of Lubich, 1983-1986, and the surveys [Bak,1982 ] and [Bru,1982].
These works also contain results of analytical interest. In some sense,
a strict separation between analysis, applications and methods of
computation is pointless.
7.2. Linear 'Vfcel Integral Equations of Second Kind
We consider, in the space C[0,a] of real or complex functions,
continuous in the compact interval [0,a] @<a<«>), integral equations of
the form

130
, x ,
G.2.1) u(x) = g(x) + jj-y J (x-t)a u(t)dt, 0<x<a ,
( ' 0
and ask for ex-L&tence and ixnA.que.ne.ii of a solution u£C[0,a].
Theorem 7.2.1: Awume 0.<a <■», A £ IR , g £ C[0,a] .
Then the -Lntegial equation G.2.1) hat, exactly one iolut-ion u£C[0,a]
In the proof of this theorem,the following lemma will be helpful
(see [Ab-St,1972, p. 258]).
Lemma 7.2.1: Von. x > 0, a>0, B > 0 we have
i-i o n, X , 4-.a-1 J ,. a + 6 D,0 .. . a + 6 r(B+1)T(a)
G.2.2) J (x-t) t dt = x BF+1,a) = x —■■— — -•-
0 r(a+6+1)
Heie B denotes the Beta-function.
Proof of Theorem 7.2.1: Introducing the operator A = XJ :
C[0,a] -» C[0,a], explicitly given by
\ x -1
G.2.3) (Au) (x) = j^-p- J (x-t)a u(t)dt, 0<x<a ,
11 0
we can write G.2.1) as
G.2.4) (I-A) u = g .
Considering Theorem 5.1.2 we now see that it suffices to show that for
oo
every f £C[0,a] the series I Anf converges in C[0,a].
n=0
The unique solution u of G.2.4) then is given as
oo
G.2.5) u = I Ang .
n = 0
By induction we shall obtain, for any f £C[0,a] and 0 <x <a, the
estimate
G.2.6) I (Anf) (x) I < II f llro \X\n r(na + 1) ^° •
G.2.6) is trivially true for n = 0. Assuming it proved for n=m >0
we find, using Lemma 7.2.1,
I (Am+1f) (x) I < 14£f J (x-fc) a_1 I ^f> «fc) ldt
1 ' 0
II £ IIJMm x
* rfeff r(ma + D I (x-fc) fc dt
l|£|lo° IXI (m+1)a r(ma+1)r(a)
r(a)r(ma + 1) r((m+1)a + 1 ]
hence G.2.6) for n = m+1.

131
From G.2.6) we conclude
oo
and with the convergence of ^ c , the proof of the theorem is completed.
Theorem 7.2.2: Aiiume 0 < a < 1 , 0<a<°°, X e IR , g£L°°[0,a].
Then the Internal equation G.2.1) hai, exactly one solution In L [0,a].
Proof: It is sufficient to introduce some minor changes in the proof
of Theorem 7.2.1 .
It is illuminating to solve explicitly, by the method described
earlier, the "te.it equation" (compare [Ke,1982])
, x _.
G.2.8) u(x) =1 + jq-y J (x-t)a u(t)dt ,
1 ' 0
that is G.2.1) with g(x) =1 for 0<x<a .
From (I-XJa)u = 1 we find
u = I A J 1 , where
n=0
,n,7na . . . X r . na-1 , ,,_
X (J 1)(x) = r,, J (x-t) 1 dt
, n .n na
X na X x
x
na T(na) T(na+1]
hence
°° . n na
G.2.9) u(x) - z ^^TT - E<Xxa), 0<x<a ,
n=0
with the Mittag-Leffler function
oo n
G.2.10) Ea(z) = ^ n^TTy for zee .
For the theory and properties of the functions E see [Bi,1945]
Exercise: Show that for g£C[0,a] the solution u of G.2.1) can be
written as

132
G.2.11) u(x) =^- J Ea(X(x- t)a)g(t)dt.
Hint: Carry out the differentiation and compare with
u(x) = X Xn(jnag)(x) = g(x) + I — J (x - t)na n g(t)dt.
n = 0 n=1 T(na) 0
The change of summation and integration is here permitted.
Comment: G.2.11) generalizes Duhamel's principle. If g £ C [0,a]
the initial value problem
G.2.12) u' (x) =g' (x) + Xu(x) , 0 < x < a, u@) = g@) ,
is equivalent to F.2.1) with a = 1 and has the solution
u(x) = g@)eXx + J eX(x~ t)g' (t)dt
0
which, after integration by parts, can be written as
d x
G.2.13) u(x) = — \ E (A(x- t))g(t)dt.
ax Q i
Let us close with a reference to [Er-Ma-Ob-Tri, 1955] and [Fri, 1963]
(for a detailed discussion of asymptotic properties of Mittag-Leffler
functions) and [Bra-Ni-Ri, 1965] for treatment of the integral equation
G.2.1) in the particular case of rational exponent a.
7.3., Analysis-Motivated Investigations
In 1958 Dinghas gave a sufficient Nagumo-type condition for
existence and uniqueness of a continuous solution to the nonlinear
integral equation
G.3.1) y(x) = T7777T ? (x - s)a~1 f(s,y(s))ds, 0 < x < a,
(«; 0
where 0 < a < 1 (note: a = 1 is admitted).
His paper contains a wealth of results and ideas and is highly
recommended to the reader. Reinermann and Stallbohm, 1971, thirteen years
later,generalized Dinghas' main result and made the proof more
transparent by an explicit use of Edelstein's fixed point theorem published 1962,
i.e. after the appearance of Dinghas' paper. Nevertheless, Dinghas used
a compactness argument, of the same type as the one used by Edelstein.
Our presentation here will be inspired by that of Reinermann and
Stallbohm.
Theorem 7.3.1 (Edelstein's fixed point theorem):
Let Y be a non-empty methic ipa.ce. with distance function p and let
T : Y -> Y be a i> elf-mapping of Y with ph.opeh.ti.eh (i) and (ii) .
(i) p(Tu,Tv) < p(u,v) foh all u,v e Y with, u 4= v.
(ii) Voh evehy yQ £ Y zke sequence (T y ) ^

133
con.ta.Zni a convergent iubie.que.nce..
Then the itatementi (a) and (b) a.ie true.
(a) There, exiiti exactly one y £ Y withTy = y.
yQ = y &oti every yQ
(b) lim T y = y &or every y £ y.
For the proof we refer to [Ed,1962] or [Re-St,1971],the latter paper
containing a proof different from Edelstein's original proof.
Reinermann and Stallbohm consider the integral equation.
G.3.2) y(x) = g(x) + -^- J" (x-s)a_1 f (x, s,y (s) ) ds ,
la' 0
assuming f,g,y to take values in IR" , f and g given, y unknown.
Theorem 7.3.2 (Reinermann and Stallbohm): kiiumptioni and notationi:
Let N be a natural number, II • II a norm in TR , a', b, M poiitive numben,
ae @,1]. Tor any poiitive number c let C([0,c], IR ) be the ipace of,
continuoui functioni from [0,c] Into IR . Define the triangle
A = {(x,s) |0<s<x<c} .
let g c C ([0,a'], IRN ) , take R ai the itt of, vzcton y e IR for which thtre
exiiti an s £ [0,a'] with || y-g(s) || < b and take
ft = A , x R, a = min {a',(T(a+1)b/M1/a} .
a
Turthermore, let f: R -» IR be a continuoui function with propertiei
(I) and (II).
(I) sup { II f (z) II I z e R} < M .
A
(II) Tor (x,s,y ), (x,s,y?) ;R the generalized Nagumo
condition holdi:
sall f (x,s,yi)-f (x,s,y2) II < r(a + D II y 1 -y 2 11 .
Statement: Then (A) and (B) below are true.
(A) There ii exaxtly one function
y e C( [0,a] , IRN ) with
II y(x)-g(x) II < ^-Apj-j- xa j5o^.0<x<a
iatiifying the integral equation G.3.2) for 0<x<a .
(B) If yQ e C([0,a], irn ) ii iuch that
yQ@) = 0 and sup { II y (x)-g (x) II I 0<x<a} < b

134
the PX.ca.Jid -iteJiat-Lon (faoJi nEE]
G.3.3) yn(x) = g(x) + jj^y \ (x-s)a~1 f (x,s,yn_1 (s))ds
conve.Jige.6 anl^oJunly on [0,a] to the. function y o {<, (A).
Comment; Notice that the domain [0,a] of definition of the solution
y may be smaller than the domain [0,a'] of definition of the function g.
Before carryingoout the details of the proof, we outline its structure
under (a), F), (y), E), (e), E), (n). Subsequent details will be
labeled (a*), F*), etc.
(a) Define a subset ZcC([0,a], IR ) so that with
G.3.4) (Tw) (x) = g(x) + j^r-p- \ (x-s) a_1 f (x, s, w (s) )ds, 0 <x <a ,
1 ' 0
we have T: Z -»Z. The number a has so been chosen in the "Assumptions" of
our Theorem 7.3.2 that the triple (x,s,(Tw)(s)) does never fall outside
the domain of definitic
Banach space with norm
the domain of definition of the function f. Consider C([0,a], IR ) as a
N
III will = max{ II w(x) II I 0 < x < a} for w e C( [0,a] , IRN) .
F) Then we set Y = TZ and show
G.3.5) u,v£Y=*l|u(x)-v(x)ll x_a -» 0 as x-» 0.
We observe TYcY, T maps Y into itself.
(y) Ill-Ill induces a metric in Y because Y c; z c; c ( [0,a] , IR
However, we introduce a second metric in Y by
G.3.6) p(u,v) = sup{ II u (x)-v(x) II x_a I 0<x<a}
and show that condition (i) of Theorem 7.3.1 is satisfied:
G.3.7) p(Tu,Tv) < p(u,v) for all u,v £ Y with u * v.
E) We show that with respect to the norm II • 11 in IR the set
{Tw wEZ} of functions is eguicontinuous and eguibounded.
Hence, by Arzela and Ascoli, for any y £Y the sequence(T y ) c _,
nk
contains a III • HI convergent subsequence (T y ) , the sequence
° k £ IN
(n, ), c of natural numbers being strictly increasing. Now, denote by
~ "" nk
y the III-III -limit of the sequence (T y ) and observe that y£Z.
° kCl

135
(e) We show that condition (ii) of Theorem G.3.1) is fulfilled:
G.3.8) p (T y ,Ty) -> 0 as k-»<= , i.e. for any y = Ty £ Y the
nA
sequence (T y ) , contains a p-convergent subsequence with limit
Ty£ Y . Note that up to now,y and Ty may depend on the chosen initial
point y £ Z.
o
(C) Now apply Theorem 7.3.1 to deduce existence and uniqueness of
solution of the integral equation, that is (A).
(n) We show
G.3.9) u,v£Y =* |||u-v||| < aa p(u,v)
and conclude that from p-convergence follows III • III -convergence, that is(B).
Details of the proof. (a*) Put
z = {w | wec([o,a], irn ), w(o) = g@), lllw-glll < b}.
Then w £ Z implies (for 0 < x < a)
II (Tw) (x)-g(x) II < yj^j J" (x-s)a~1 ds = r("+1) xa ,
in particular (Tw)@) = g@) and the estimate
II y(x)-g(x) II < r<"+1) xa
for any solution y of G.3.2) in 0<x<a .
Furthermore
III Tw-cj Ml < r(q+1) aa < b .
Hence also Tw £Z,i.e. T maps Z into Z.
F*) To prove G.3.5),we introduce fif : [0,°°) -> ]R , the
modulus of continuity of the function f, by defining fi, E) as the
supremum of II f (x , s ,y )-f (x?, s? ,y?) || under the conditions
A
(x.,s.,y.) £ R for j = 1 and j = 2, max{ Ix^x.^, I s1 -s2 I , II y1 -y2 II } <6,
and observe the following properties:
fif is continuous at 5 = 0 because f is uniformly continuous on the compact
1 A
set R, and nf@) = 0 .
Furthermore, Q^(S) is a nodecreasing function of S .
For functions u,v£Y there are functions u,v £ Z such that u = Tu ,
v = T v . Using the properties of fi, , we conclude

136
II u(x)-v(x) II < jrrK- \ (x-s)a 1 || f (x,s,u(s))-f (x,s,v(s)) II ds
1 ' 0
a ^ .
< r ,X+1 > nf (max {||u;s)-v(s)|| 0 < s < x}) ,
hence || u(x)-v(x) II x a -» 0 as x -» 0.
(Y*) As an exercise the reader should prove that p as defined by
G.3.6) is a distance function in Y.
Now let u,v£Y, u * v. We must show that G.3.7) is true. Define
ku v : [0,a] - ]R by
0 for x = 0
k (x)
U , V
x II u (x)-v (x) || for 0<x<a ,
observe that k is continuous and conclude
u, v
p (u , v) = max {k (x) I 0 < x < a} .
u, v - -
Now either p(Tu,Tv) = 0 in which case G.3.7) is trivially true, or
p(Tu,Tv) >0. In the latter case there exists a number C£@,a] such that
p(Tu,Tv) = ra|l (Tu)(?) - (Tv) E) || ,
hence by the generalized Nagumo condition (II)
p(Tu,Tv) 5 yj^j \ (C-sH || f (S,s,u(s))-f U,s,v(s)) II ds
r
_ r-a r(a+1) f ,,. ,a-1 -a,, ,- , - ,, ,
< ? pj-y J (fc,-s) s || u(s)-v(s) II ds
< ra a \ E-3H-1 kn (s)ds
0 u,v
< a Ca p(u,v) J E-s)a~1 ds = p(u,v),
0
so G.3.7) is true.
E*)We shall check theeguicontinuity and eguiboundedness of the
functions Tw for w e Z. Let w £ Z and 0 < x < x_ < a .
Then
II (Tw) (x2)-(Tw) (x^ II < Iq t i + i2 + i3 wiLh (see (I))
lQ = II g(x2)-g(x ) II ,

137
'1 r(a) J
i \o-1 , ,0-1
(x2-s) - (x -s)
llf (x,,s,w(s) ) II ds
a , a a.
r(
^TTT {|x2-xl' " (x2 " X1)} '
I2 = YT^Y J" (^^5H1 1 llf (x2,s,w(s))-f (Xl,s,w(s)) || ds
o,£ (ix -Xp i;
r(a+i) A1 '
I3 = rbo J" (x2"sH( 1 " f (x2,s,w(s)) II ds
x1
M
r(a+l) VA2 A1
(x0-x,)
Putting together these estimates we obtain
G.3.10) II (Tw) (x2)-(Tw) (Xl) II <
< II g(x,)-g(x.) II +
2' yiAi' " T r(a + 1) L"A2 Ai
{2|x0-x,Ia + lx"-x?|}
2A1n r(a + l) fVl 2 Ai
fif ( lx0-x. I )
which is, by symmetry, valid for all x ,x? £ [0,a] regardless whether or
not x_ > x , and gives the desired equicontinuity.
Equiboundedness follows (see(a)) from
II (Tw) (x) II < II (Tw) (x)-g(x) II + II g(x) II < b + III g III .
(e*) We shall prove G.3.8). Let y £Y and T y-»yCZask-»<=
in the norm III • HI . See E) . Then for 0 < x < a and k e 3N we have
-a nk+1
x a||(T K yQ) (x) - (Ty) (x) II
< — x~a \ (x-s)a 1 II f (x,s, (T y) (s))-f (x,s,y(s)) II ds
r(a) 0
X
a-1
< j^y X'" J" (X-S)a ' dS • fif (INT Kyo-?lll)
and from E) and the properties of fi, we deduce

138
nk-M n
p(T y0-T?» <- f^ttt nf (II|T y0-?m»
-» 0 as k -» oo f
that is G.3.8).
(£*) The conditions of Theorem 7.3.1 satisfied, we conclude that
in Y there exists exactly one y with Ty = y and for any starting point
y c Y the sequence (T y ) r ._T converges to y in the metric defined by
the distance function p. Because TwZ Y for any w £Z we can even take
y EZ and have the same convergence property.
(n*) To demonstrate G.3.9) deduce the chain of equalities and
inequalities
III u-v III = max II u (x) -v (x) II = sup ||u(x)-v(x)ll <
0<x<a 0<x<a
< sup x II u (x) -v (x) II a = a p(u,v).
0<x<a
The long proof of Theorem 7.3.2 is thus completed. We close this
paragraph with the presentation of Dinghas1 counter-example exhibiting r (a +1]
as the optimal constant in the generalized Nagumo condition (II).
To this end^consider the function u(x)
for x > 1 and calculate uA) = 1 and u'A\
T(a+x)
r(a+1)T(x)
r1 (a + 1) _ I" A)
r(a + i) rd)
The strict logarithmic convexity of the gamma function for positive
argument implies u'A) >0, and we see that for sufficiently small positive e
there exists a positive solution 8 = 6(e) of the equation uA+8) = 1 + e
with the property B(e) -»0 as e -» 0.
Now define, with 8 = 8(e), the function
rr(a + 1) (l+e)s"ay for 0<y<sa + 6
f(s,y)=J„, ,. , „ ,8 , a+8
11 ' J T(a + 1)(l+e)s for y>s
0 for y < 0 .
Then for any c£ @,1] the function y(x) = c x , x>0, solves the
integral equation G.3.1), namely

139
G.3.11) y(x) = YT^Y $ <x-s>a 1 f<s'y (s))ds, x>0,
0
which thus has infinitely many solutions.
In fact, writing
G.3.12) z(x) = y\^) Kx-S)a 1 f(s,y(s))ds ,
we find
x
/l
0
z(x) = -p-f-^y J (x-s) f (s,cs )ds .
0
r(a)
that
Since c sa+6 < sa + 6, it follows (use Lemma 7.2.1)
x „
z(x) = aA+e) \ (x-s)a 1 c s ds =
0
= caA+e)xa^ HB+nna) , cA+£)x«+B r(a + l)r(B + i) ^
r(a + 6 + D T(a + 6 + i)
But u A+6) = 1 + e means at |f —=y = 1 + e, and we see that z(x) = y(x).
Looking back at G.3.12) shows that G.3.11) has infinitely many
solutions.
7.4. Applications-Motivated Investigations:
Problem Formulations, Newton's Law of Cooling
In 1951 Mann and Wolf published their pioneering paper treating heat
conduction on a half-line x > 0 with initial condition given at time t = 0
and a nonlinear radiation condition at the boundary x = 0, t>0. They
reduced this problem to a nonlinear Abel integral equation on the
boundary x = 0, t > 0.
Their results were later generalized in various ways (see [Ro-Ma,1951],
[Pa, 1958], [Le,1960], [Ke-01,1972] , [Ol-Ha, 1 976 ] , we shall give a review of
tnese works in the next paraqraph). Furthermore,numerical methods have been
developed for approximation of solutions (see [Ba,1982], [Bru,1982], [Gr,1982],
[Gr,l985], [Ha-Lu,1986], [Ke,1982], [Li,1985], [Li,1969],
[Lu,1983], [Lu1,1985], [Lu2,1985], [Lu1,1986], [Lu2,1986]). For a general
orientation and a treatment of specific questions, we refer to [Mi ,1971],
[Mi-Fe, 1971] , [01-Sp,19 74], [Ca,1984].
In order to have at hand a general theorem on existence and
uniqueness of solution of the Neumann initial-boundary value problem (N) for
the heat equation in the quarter plane x>0,t>0, we condense Theorems
5.2.1 and 5.2.2 of Cannon's book [Ca,1984] into our Theorem 7.4.1 .

140
(N) Determine u(x,t) « x>0, t>0 i,o that
ut = uxx ^ofl x > °' fc >0'
ux@,t) = g(t) ion t >0,
u(x,0) = f(x) ion x > 0.
I i f acid g a/te continuous, toe /teqa-i/te
lim u (x,t) = g(t) (Jo-I t>0,
x-0 x
lim u(x,t) = f(x) hoK x>0.
t->0
We shall use the kernel functions (Green's functions)
2
G.4.1) K(x,t) = — exp (- f-r), xEI, t>0,
\/4TTt
G.4.2) N(x,5,t) = K(x-5,t) + K(x + 5,t), x, £ e IR , t>0 .
Theorem 7.4.1 Le-t -tfie (,unctioni g(t) (Jo/i. t>0 a fid f (x) (Jo/i. x>0
be continuous* and tut f w-t-C/i Au.A.£a.bte. conhtanti, C ,C.£[0,°°) and a £[0,1]
Aatii&y a gtiowth condition
If(x)| < C1 exp(C2 x1+a) .
Then the function
o° t
G.4.3) u(x,t) = \ N(x,5,t)fE)d5 - 2 J" K(x,t-i)g(t)dx
0 0
^oa. x > 0, t>0 ii, iolution o{, the Neumann problem (N) .
This, hotution it, unique within the ttaht, of, hotutioni, v iatii^ying with
nonnegative conitanti C-. and C. a gfioujth condition
Iv(x,t)I < C3 exp(C4x").
One now arrives at an Abel equation of second kind if at the
boundary x = 0, t>0 instead of the values g(t) of u @,t) a radiation
condition
G.4.4) u @,t) = F(t,u @,t) ) , t>0,
is prescribed, connecting the outward flux u @,t) (or the inward flux
-u @,t)) with the boundary temperature u@,t).
Remark: We imagine here the following situation to be modelled:
u(x,t) denotes density of heat (which is energy) and simultaneously
(by proper choice of units and zero-temperature point) temperature.

141
However, we could also imagine u (x, t)as the density of a diffusing material
substance.
Putting
G.4.5) ip(t) = u@,t) , t > 0,
G.4.6) g(t) = F(t,ip(t) ), t > 0,
and inserting into formula G.4.3) of Theorem 7.4.1,we obtain
o° t
ip(t) = J" N@,5,t)f E)d5 - 2 J" K@,t-T)F(T,ip(T))dT
0 0
which with x = 0 in G.4.1) and G.4.2) reveals itself as an Abel
integral equation o& second kind
oo ? t
G.4.7) vit) = -i- \ exp(- |l)f ((-)d? - -±- \ Ull^llU. dT
Vwt 0 \/tt 0 \/t-T
for the determination of the unknown function ip (t) = u@,t) whose insertion
via G.4.6) into G.4.3) gives us u(x,t) in the whole quarter-plane x>0,
t>0 if tp(t) is continuous in t>0.
Remark: G.,4.7) it, a nonlinear integral equation H, F(t,y) de.pe.ndi
nonlineafily on y. Otherwise G.4.7) is a linear integral equation.
Alternatively for the determination of u(x,t), the solution formula for
the Dirichlet problem (D) may be used.
(D) Determine u(x,t) i» x>0, t>0 io that
ut = u loh. x > 0, t > 0,
t xx u
u @, t) = ip(t) iofi t > 0,
u(x,0) = f(x) ^01 x > 0 .
1M f and ip are continuous vie require
lim u(x,t) = tp(t) faoK t>0
x->0
lim u(x,t) = f (x) {<,or x>0
t-»0
The solution formula is (see Chapter 4 of Cannon's book for the details
and conditions of validity)

142
G.4.8) u(x,t) =-2/|^ (x,t-x)ip(T)dT
0 dx
oo
+ \ G(x,£,t)fU)d£
0
with
G(x,5,t) = K(x-5,t)-K(x+5,t) .
Let us now treat in detail the problems of Newtonian heating.
The more customary problem of Ne.wton.ian coot-ing can be treated analogously:
at appropriate places signs have to be inverted.
The problem of heating has also briefly been discussed in [Ma-Wo,1951].
Consider a semi-infinite rod (x > 0) in which heat conduction is taking
place-, to the left of which there is constant temperature 1. Newton
assumes radiation at the boundary x = 0, t>0 to be proportional to the
difference of outside temperature 1 and inside boundary temperature
u@,t), that is - u @,t) = cA-u@,t)) with a nonnegative constant c.
For simplicity we assume, following Mann and Wolf, the initial temperature
u(x,0) to vanish. Hence we are faced with the following problem.
(NH) Vetetimlne u(x,t) a x>0, t>0 io that
ut = uxx ^ofl x >0' fc > °'
- u @,t) = c(l-u@,t)) io-i t>0
with a given con&tant c>0,
u (x,0) = 0 ^OK x > 0 ,
and ^uKthefimofie
lim u(x,t) = 0 ^oh. x>0,
t->0
lim u(x,t) exi-hth and i.i> contlnuoui ^ofi t>0 .
x->0
Inserting F(t,z) = -c(i-z) and fE) = 0 into G.4.7), we arrive at
the JLineati iecond kind Abet Lntegftat equation
o t ( \
G.4.9) tp(t) = i£ \/t - -0- / ^-111 dT , t>0
Vtt \Jt\ 0 Vt-T
If (NH) is a good model for the (Newtonian) process of heating from
outside by inward radiation proportional to the difference of temperatures ,
the solution of G.4.9) should reflect properties of the physical process

143
which can be observed and which one expects intuitively. Namely: tp(t),
the inside temperature at the boundary x = 0, should be a continuous
function strictly increasing from ip@) = 0 towards lim tp(t) = 1
t-*»
if c>0. In the trivial case c = 0 we should have tp(t) = 0 for all t > 0.
In order to show that ip does indeed behave in this way,we explicitly
solve the integral equation G.4.9) using the Laplace transform method.
D.etails are left to the reader whom we assume to be familiar with this
technique. The result is (see[Ab-St,1S72])
G.4.10) ip(t) = ^ Vt - c2 J exp(c2s)erfc (cv's)ds
\/tt 0
where
2 °° 2
G.4.11) erfc (r) = -j- \ exp(-s")ds, r e IR ,
\/tt r
is the compZe.me.nta.fLy ennoi &unc£J.on.
Obviously tp(t) is continuous for t > 0, ip@) = 0, and in the trivial
case c = 0 we have tp(t) = 0 for all t>0 . But if c > 0, which we henceforth
will assume, the global behaviour of ip cannot be seen immediately.
We can simplify by getting rid of the constant c. Substituting
r = c\/s, we get
? _ c\/t ?
tp(t) = ^- Vt - 2 J" r exp (r ") erf c (r) dr,
\/tt 0
and by a second substitution s = c\/t, we find
' ip(t)
G.4.12) <
<Ms)
for t>0 or s>0, respectively.
For an investigation of the growth properties of ty, we differentiate
2 2
ty' (s) = -^- - 2 s exp(s )erfc(s)
= -^ A -2s exp(s") \ exp(-r')dr).
Vtt s
= *(c\/t)
2 s 2
= -^ s - 2 \ r exp(r ")erfc(r)dr
\/tt 0

144
By the inequality G.1.13) of [Ab-St,l972] we have
2 °° 2 1
exp(s") \ exp(-r")dr < —=^_ for s>0 ,
/24
+ vs + —
IT
hence
ty' (s) > — A- 2 s
V% /TT
s + vs +—
G.4.13) ip'(s)>0 for s > 0.
It follows that ty[s) and tp(t) are strictly increasing for s>0 and t>0
respectively .
Therefore tp(t) tends to a limit y £ @,°°]. To show that y = 1 we
assume the contrary. There are two cases;
(i) 1 < y < + °° , (ii) 0 < y < 1
In case (i) there exist numbers be A,°°) and t £ @,°°) such that
t>to=»ip(t) >b. Let t>t and deduce from the the integral equation
G.4.3) (taking account of tp(t) >0 for t > 0)
, t
ip(t) < ^ Vt - 4: \ —— dT
VV \JTi t Vt-T
2c , /-
(\/t - b \/t-t ) -» - °° as t->°°
,— o
VTT
which is not compatible with (i).
In case (ii) we have 0 < ip (t) < y < 1 for all t>0, and G.4.9)
implies
ip(t) > i£ Vt - -% / —— dT
\/tt \/tt 0 Vt-T
= ^- A-y) v't -» °° as t-»°°,
which again contradicts the assumption.
We now derive by the infinite series technique described in § 7.2
another representation of the solution ip of G.4.9), namely a rapidly
converging infinite series which is, in some sense, complementary to the
integral representation G.4.10).

145
By formula G.2.11) and G.2.10) we have
ip(t) = A J" E. ., (-c(t-sI/2) % Vi ds
aC 0 '' \/tt
n j t °° . „.n n . .n/2 ._
= 2£ d ; z (-1) c (t-s) s1/2 ds
V% 0 n = 0 r E + 1 )
2c d ™ (-1)ncn r /4- *n/2 1/2 ,
= — -3-r I ! J (t-s) ' s ' ds .
\/tt n=0 rB + 1) o
Convergence is so fast that summation and integration can be interchanged.
Now application of Lemma 7.2.1 yields
*{t) =2£ A " (^)nf r<3/2) J+^ .
v/¥ dt n=o r (§ + |)
Thanks to fast convergence we can interchange summation and
differentiation, and using \/tt = FM/2) we obtain
n 1
G.4.14) ip(t) = X J—L> 2 t
n=0 r(|+|)
= 1-E1,2 (-c \Zt) for t>0
and correspondingly, with s = c \/t > 0 ,
G.4.15) i|i(s) = X ' '' sn + 1
n=0 r(£+|)
= 1 -E1/2 (-s) .
We collect the results as a theorem.
Theorem 7.4.2; The integral equation G.4.9) hat, at, notation the
boundary value u@,ti = tp(t) o<5 the heating, pfioblem (NH) , and we have
tp(t) = ^°- \j1 - c'" \ exp (c"s) erfc (c \/s) ds
\/tt 0
= 1 - E1 /2 (-c\/t) ioK t > 0.
Tfie (Juiac-C-toia tp(t) it, continuous and bth.lc.tly lncfieat>i.nQ {^ofi t>0, and

146
— ?c
ip@) = 0, ip(t)/\/t -» ^ a* t-»0, ip(t) -» 1 ai t-»<= .
Remarks: The partial sums
G.4.16) <a(t) = z i_Ii_°_ t- , kEI ,
K _ „ ,n J. o
n=0 r <2 + 2*
of the infinite series in G.4.14) for ip(t) can be obtained by a Picard
iteration applied to G.4.9) according to
tPQ(t) = 0
G.4.17) J , ,
ipk (t) = — Vt - -0- / K ' — dT , ken.
\At \/TI 0 Vt-T
Again Lemma 7.2.1 x:an be used fcdjaaiculate the integral on the right-
hand side. This iteration method has been generalized and successfully
applied to the nonlinear problem by [Ma-Wo,1951].
The limit relation ip (t) /\/t -» 2c/\/tv as t -» 0 can also be seen from
the series representation G.4.14).
7.5. Applications - Motivated Investigations;
Survey of Literature
We shall give an overview of important contributions with selected
results published since 195 1 on the Kadlatlon~dl((u(,lon pKobtem described
in § 7.4 where we have treated the linear case. For the reader's
convenience,we reformulate the problem.
(RD) VeteKmine u(x,t) (oK x>0, t>0
io that
ut = uxx &ofl x > °' t > °'
u(x,0) = f(x) (ok x > 0 ,
ux@,t) = F(t , u@,t)) (OK t>0 .
1( f and F a.Ke. conti-nuouA, we. Ke.qui.Ki that
lim u (x,t) = F(t,u@,t)) (ok t>0 ,
x-0 X
lim u(x,t) = f(x) (ok x>0 .
t-»0

147
To have a correct visualization of what is happening,consider
u(x,t)as the densityof an extensive quantity (of a substance or of energy,
for example) distributed along the positive halfline and having at x = 0
inward flux -u @,t) at time t. From Theorem 7.4.1 we can deduce that the
x
problem has a solution u(x,t) given by formula G.4.8) if the conditions
(i) , (ii) and (iii) Jbelow are met.
(i) F li a contlnuoui function on [0,°°) x ]R ,
(ii) the aaoclated Integral equation G.4.7) na.me.ty
o° 2
G.5.1) ip(t) = -4= \ exp(- 4r)f(£)d£
V-rrt 0
_ _L ; f(t^t)) d.[r t>0 f
\/TT 0 Vt-T
hai a unique contlnuoui iotutlon,
(iii) f iatli^lei the gtiowth condition o {, TkeoKem 7.4.1 .
This solution is unique within the class of solutions v satisfying the
growth condition
Iv (x,t) | < C3 exp (C4x").
In the cases to be listed below we , we simply have to specify the functions
f and F or describe their assumed general properties.
In 1951 Mann and Wolf assumed f(x) = 0 for 0 <x < °° and F(t,y)=-G(y)
for y e IR . They further assumed | u (x, t) | <M for 0 < x < °°, 0 < t < °°, an
assumption which we know can be considerably relaxed (see § 7.4).
First they discussed (not in such detail as we did in § 7.4) the
linear problem of outside density (in their context temperature) s 1 and
Newton's linear radiation condition leading to G(y) = cA-y) with a
positive constant c. Then by abstraction to a more general radiation
condition, still assuming outside density = 1 for all times t, they formulated
their essential hypotheses (A),(B),(C).
(A) G(y) li continuous f^oK. - °° < y < °° .
(B) GA) = 0 .
*)
(C) G(y) li itfilctly decKeahlng In y.
They arrive at the integral equation
G.5.2) ,p(t) = 4; ; GiiilU dT
\/TT 0 \/t-T
*) They say "monotone decreasing" but by the way they draw conclusions one sees that
they mean "strictly decreasing".

148
and show, using Schauder's fixed point theorem, that under the
assumptions (A) , (B) , (C) for any bounded interval 0<t<T there exists
at least one continuous solution <p(t) of G.5.2) satisfying tp(O) = 0
and
G.5.3) 0<tp(t)<1 for 0<t<T .
Under the additional assumption that G(y) satisfies a Lipschitz
condition in 0<y< 1 they show that the mapping
B : C[0,T] -» C[0,T]
defined by
(Bv)(t) =-L ; g*(v(t)) dT
n/tt 0 \/t-T
where G*(y) = G(y) if y<1, = 0 if y>1, generates from ip = 0
a sequence of iterates ip = Btp with all ip @) = 0 and for t>0 having
the alternating monotonicity property
tPQ(t) < tp2 (t) < tp4 (t)< < tp5 (t) < tp3 (t) < ip1 (t) .
They also show the existence of a continuous limit function
ip(t) = lim ip (t) ,the convergence being uniform on any bounded interval [0,T],
n-*»
which is a solution to the integral equation G.5.2) with the property
0 < ip(t) < 1 for t > 0.
Introducing more assumptions on the function G they deduce further
(intuitively expected) properties of ip. We suggest that the reader look
into Mann and Wolf's excellent exposition and content ourselves with
reporting two other important results.
If, in addition to (A) , (B) , (C) , the function G is LZpic.hi.tz-contln-
uoui on [0,1 + e] fioA. a. po&itive. e, then ip(t) is non-decreasing for t>0,
and 0 < ip(t) < 1 for all t>0, tp(t) -> 1 for t -»=°.
-1/2
In the same year 195 1 Roberts and Mann replaced (t-t) in the
integral equation G.5.2) by a function K(t-i) reflecting the essential
-1/2
properties of (t-i) . They thus could extend the theory to more
general integral equations.
In 1958 Padmavally investigated the problem (RD) and the integral
equation G.5.1) with f(x) = 0 for 0 < x < °°, F(t,y) = -g (y, <j> (t) ) .
She denoted the outside density (or temperature) by § (t) and postulated the
following natural properties for the flux g .

149
(a) g(u,v) continuous. In both u and v.
(8) g(u,v) ittiictly dectieaiing In u f,oti fiixed v,
ittiictly inctieaiing .in v faofi {iixe.d u.
(y) g(u,u) =0 ^oK all u.
To understand the significance of these properties,take into account
that u corresponds to the density (or temperature) at the endpoint x=0
of the half-line x>0 , and v to the outside density (or temperature)
just to the left of x = 0.
The data of Padmavally's problem consist of the functions g and ¢.
She furthermore assumes that the discontinuities of <j>(t) in t>0 form
a discrete set and ¢ (t) is bounded in any finite interval 0 < t <T. Her
integral equation is
G.5.4) <p(t) = 4; ; g(tp(T^(T)) dT ,
\Jt\ 0 \Jt-x
and by a lot of hard analysis she obtains the following results
(intuitively expected).
(a) !{<, <)> (t) < <)>?(t) iofi 0<t<T and u (x,t), u2(x,t) atie the
cotitieiponding iolutioni o{. the. tia.dia.tJLan di&&u&ion ptioblem (RD)
with f(x) = 0 and F(t,u@,t)) = -g(u@,t), cf> (t) ) ushene $ = ^,
$ - <)>.,, tie.ipe.ctive.jty, then u (x,t) < u?(x,t) faoti x>o, t>0.
(b) I;; m<0, M>0 a.tie. conitanti iuch .that m<<J>(t) <M f^oti 0<t<T,
then m<u(x,t) <M {-on 0<t<T, x>0 .
(c) 1E ¢@) > 0 and <)>(t) ii non- dectieaiing then ip(t) = u@,t) i&
alio non-dlctieaiing, 0< tp(t) < $ (t) faoh. all t>0 ,
ip(t) <<ji(t) ioti thoie t ioti which <))(t) >0.
(d) 1E X = lim <)> (t) exiiti and ii finite, then tp(t) -» X ai t -»°° .
t-*»
In 1960 Levinson, motivated by a problem of superfluidity theory,
took problem (RD) with
f(x) = 0 for x > 0,
F(t,y) = .My-f'tt) ) , t>0, -oo<y<oo.
The data are the functions $ and f, assumed to be continuous. It is
further assumed that <t> is strictly increasing and ¢@) = 0. This is a

150
special case of Padmavally's problem. Levinson is interested in the
particular situation where f"(t), the outside density (or temperature), is
peAiodic.
His integral equation is
G.5.5) ,p(t) =-4;/ <H<p(t)j-?(t)) dT > t>0 .
\/tt 0 \/t-T
With ip = ip - ? it is equivalent to
G.5.6) 5(t) + ?(t) =-4/ 1»(ip_m) dT .
\/TT 0 Vt-T
One may expect that as t grows and grows the solution tp(t)becomes
closer and closer to a periodic function. Levinson proves two theorems.
(A) Let f"(t) be continuous (,oa 0 < t < °° and t>atit>(y a uni(,oAm Holde.fi
condition*} o {j oA.de.fi R > 0 on any finite inteAval . LeX<J>(y) be ttfuxXZy
incAeating, ¢@) = 0, and &oa any y >0 let theAe exitt a constant
K(yQ) tuch that l<My2)-<My.,) I < K(yQ) 172-7-,1 iofl lY-,1 < YQ>
I y _ i < y . Then G.5.5) poi>i>ei>i>ei> a unique continuous, solution tp(t)
(,oa 0 < t < °o.
(B) In addition to the hypothetet, o & (A) at>t>ume that f"(t) hat, peAiod u
and that with M = max | f" (t) | theAe it, a positive ttAictly incAeaiing
function k(u) (on u>0 iuch that <$> (y2 ) -<t> (Y1) > k(y2~y ) (on
y2-y1>0 and ly^ < 2M, |y2l < 2M .
Then theAe it, a continuous peAiodic function tp*(t) o^ peAiod u
Aucfi -Cfia-C |ip(t)-ip* (t) I -» 0 a-6 t -»<= .
Moneoven |ip(t)| < max|f"(t)| (oa t>0 .
Starting from 1972 , Keller, Olmstead and Handelsman published a
series of papers in the following years of which we quote [Ke-Ol,1972]
and[01-Ha,1976]. In [Ke-01,1972] problem (RD) is taken up with f(x) = 0
for x>0 and F(t,y) = a yn-f"(t) for t>0 where a>0 is a constant. The
data are the function f" (the outside density or temperature), the
constant a and the exponent n, the latter assumed to be positive.
In [Ol-Ha,1976] f need not be the zero-function and, more generally
than in [Ke-Ol,197 2] ,
F(t,y) - G(y) - f (t) , t >0, y£l,
*) Levinson uses the words "Lipschitz condition"

151
The data are the functions f,G,f" .
In [Ke-01,1972] as well as in [Ol-Ha,1976] it is assumed that u(x,t)-»0
as x -»°°, a condition which can be considerably relaxed (see the beginning
of this paragraph and the quotations from Cannon's book in the preceding
§ 7.4) . The main interest of Keller, Handelsman and Olmstead is the
asymptotic behaviour as t -»oo or t-» 0 in the dependence on the exponent n or on
the asymptotic behaviour of the function G . We do not reproduce the
extensive tables they have calculated.
In [Ol-Ha, 1 976 ] a proof is given for the existence and uniqueness of a non-
negative solution ip(t) under the hypotheses ('), ("), ('") •
(') G(y) li contlnu.ou.Aty dl^eKentlable iofi y>0 and kai a well-defined
~ 1
ln.ve.Jiie. function G (Y) .
(") G@) = 0, 0<G'(y) < y &oK 0 < y < yQ ,
whefie y >0 may depend on y
('" ) Tke function
g(t) = f(t) + - \ f(?) ? exp(- fr)dC , t>0 ,
2v/nt3/2 °
li locally intestable and 0<g(t) <M , with a conitant M.
A sufficient condition for (''') to hold is that f(t)/t and f(t) are
non-negative, bounded and locally Integfiable.
The proof is done by applying compactness arguments to a sequence
of Picard iterates.
Another topic dealt with is the asymptotics (as t-»«>) of
t
E(t) = \ (g.(s)-G(ip(s)))ds ,
0
which in the particular case f(x) s o is the net inward flux
across the boundary x = 0 .
Final remark: The applications-motivated investigations reported
here are, in the way they are carried out, completely independent from
the analysis-oriented ones described in 7.3. The conditions concerning
the nonlinearities are very distinct from each other.

152
7.6, A Very Brief Survey of Literature on Numerical Methods
In 1982 and 1985,Groetsch treated numerically the problem, posed
in 1951 by Mann andlWolf, He analyzed and tested an equidistant piece-
wise linear ansatz for an aoproxiir.ation to the solution tp(t). He showed
that if ip£C"[0,T] where T > 0 is arbitrary (but fixed) and G (see 7.5)
satisfies a Lipschitz condition with constant L< 1/\/ttT then the
approximate soluticasconverge uniformly to the exact solution in each
interval [T ,T] provided T > 0. Convergence is of the order of the steplength
h. His bound 1/\/ttT for the Lipschitz constant is very restrictive and
further investigations whether this restriction can be removed or
relaxed are desirable.
In 1969 Linz analyzed product integration methods for integral
equations of the form
^ x
u(x) = g(x) + J" p(x,t)K(x,t,u (t) )dt ,
0
under the essential condition that K is continuous and in particular
Lipschitz continuous with respect to its third argument u.
-1/2 7 7 -1/7
Typical forms of p(x,t) are (x-t) and t(x'-t')
In 1982 Kershaw treated by the product trapezoidal rule integral
equations
u(x) = a(x) + ^- if K(x't'u(t)) dt
u(x) g(x) r(a) j ^^^^ dt,
where 0< a < 1 and K satisfies certain conditions. He also gives a
theorem on the existence of a solution by applying Banach's fixed point
principle.
J.J. te Riele in 1982 described a
imating solutions of equations of type
X —1/2
u(x) = g(x) + \ K(t,u(t))(x-t) /- dt
0
1 / 2
where u (x) = x(x) + x il) (x) with smooth functions x an^ i>•
He thus took specific consideration of possible non-differentiability
of the solution u(x) at the origin x = 0.
Linz, Kershaw and te Riele considered their integral equations in a
finite interval 0 < x < a .
In very recent years several contributions to the treatment of
linear and nonlinear Abel integral equations by multistep methods and
by methods of Runge-Kutta type have been given by Hairer, Lubich and
Schlichte. See [Ha-Lu , 1 986 ] , [Lu,1983], [Lu1,1985] ,
2 1 2
[Lu",1985], [Lu ,1986], [Lu",1986]. They have succeededin generalizing

153
the Dahlguist theory of the special Volterra integral equation.
x
u(x) = u@) + J f(t,u(t))dt,
0
corresponding to the limiting case a = 1 of Abel's integral equation.
A-stability and related concepts are studied by Lubich in particular for
the linear test equation
, t
u(t) = f(t) + jj^y J" (t-s)a u(s)ds, t>0,
where 0 < a < 1 .
The reader interested in numerical methods should consult the
monograph of Brunner and van der Houwen A986) .

Chapter 8: Illposedness and Stabilization of
Linear Abel Integral Equations of First Kind
8.1. General Topics in Ill-Posed Problems
In Chapter 6 we observed that the problem of inverting the Abel
operator is ill^-posed. The inverse of the operator exists, but is not con-
tinuous in L"-norm. Here , we develop a more general notion of
illposedness and discuss other examples. For our aims,it is less important
to give a formal definition; for these, we refer to the bibliography.
We reformulate Hadawrd's definition of well-posedness in the context
of linear mappings between normed linear spaces.
Let X and Y be two linear normed spaces and let A: X -» Y be a
linear operator. We say that the problem of solving the equation
( 8.1.1) Au = f
where f £ Y is given and u £ x is unknown,is well-posed if the following
three conditions hold.
(i) The equation (8.1.1) has at least one solution for general data
f (that is the operator A is surjective).
(ii) The equation (8.1.1) has at most one solution (that is A is in-
jective).
(iii) The solution u of (8.1.1) depends continuously on the right-hand
side f (that is A : Y -» X is a continuous operator).
We say that the problem (8.1.1) is ill-posed if it is not well-posed.
There are many physical problems that,in mathematical formulation,
are ill-posed.
When condition (i) is not satisfied,the space Y is "too large",
in other words, the data of the problem are incompatible. This is the
case, for example, for the simple linear system
f X1 + x2 = 1
(8.1.2) < x1 - x2 = 0
,2x1 + x2 = 3
2 3 2 3
that has no solution. Here X = IR" , Y = IR , and A : IR" -* IR is
defined by

155
A(x ,x2)
In some sense,there are "too many data" for having existence, i.e., Y
is too large.
When condition (ii) is not satisfied there are, often, too few data
for the unique determination of the solution. Consider, for example,
the linear system
x + x? + x. = 1
x - x2 + x3
Here X
IR~
IR"
and A: ST
IR"
is defined by
A (x , x« , x~)
1 1
1 -1
and the system does not have a unique solution. The space Y is too small.
When condition (iii) is not satisfied,we have, generally, a more
pathological situation. Then it may happen that, when the data are known
up to a small error,the solution can be determined only with a very large
error. It is impossible to treat a physical problem if its formulation
doesn't satisfy (iii). In fact, in a practical problem,the data are never
known accurately,therefore we cannot extract any useful information about
the solution.
The problem of inverting the Abel operator in Lp@,1) spaces doesn't
fulfil the condition (iii). Tc prove this we use the compactness results
proved in 4.3. The Abel operator Ja: LP@,1) -» LP@,1) is,by the formula
io •, ?\ -ra i \ 1 r u(t)dt
(8.1.3) Ju(x)= r ,„, J '
Ha)
0 (x-t)
1-a
a compact operator, therefore (J
•1
cannot be continuous. If it were
continuous then Id = (J ) (J ) would be compact (see [Ta-La, 1968;
theorem 7.2, p.298]),that is the unit sphere in Lp@,1) would be compact.
But this cannot be the case because Lp@,1) is not of finite dimension (see[Ta-
La, 1968; th. 3.6, pag. 65]).
A direct proof of the non-continuity of the inverse (J
by the following example.
■1
is given
Example 8.1.1: With
sin irnx
fn(x)
Un)
a
0 < x < 1 ,

156
we have
and
rrnx
un(x) = ,jV fn(xj = r^ I cos(—"g) d?
II f II -» 0 as n -» °° ,
n L~@,1)
lim Mull > - .
n-w n l'(o,D ~ ^
It follows that (Ja) is not continuous from l"@,1) to Lp@,1). In fact,
by an application of Holder's inequality
lim II (J0)-1 f || > limll (J0)-1 f II .
n-«*> Lp@,1) n-«> l'@,1)
= lim ||u || > 1 .
n-«» "' l1 @,1) n
In the study of an ill-posed problem a very important question
arises: How to restore stability (i.e. continuous dependence on the data)
in a problem not satisfying the condition (iii)?
When (8.1.1) is the mathematical formulation of a physical problem,
we must be careful to use all informations that the physical situation
can suggest us. For example the sign of solutions, see [Pu,1959], the
boundedness of energy, [Pa,1975] or of some derivatives of the solution,
the monotonlclty or convexity properties of the solution, and so on. These
types of a priori informations are generally available directly from the
physical problem and, in some sense, they are "more true" than the
equation of the problem (see Pucci [1959,§ 3]). In many cases, the a
priori informations on the solutions guarantee that the set of possible
solutions becomes a compact subset of the space X. This is a happy
situation, in particular, when A: X -» Y is an invertible and continuous
operator. In fact in this case, by a theorem of general topology (see Tlkhonov-
Arsenin [1977]) , the inverse of A|„ , where K is a compact set in X, is
a continuous map from A(K) to K. In any case, even if A is a continuous
invertible operator and K is a compact set in X, it is very important
to give a precise evaluation of the modulus of continuity of the operator
_ 1
(Al ) , i.e. to find as good as possible stability estimates for the
solution (we illustrate this in an example below)
The final step, crucial for the applications, consists in finding
a constructive method to approximate the solutions of the problem. We
consider this question in Chapter 9.

157
Example 8.1.2; Let X = C°([0,1]), Y = {u £ C1([0,1]), u@) = 0}.
X and Y are normed linear spaces with the sup-norm. Let A be defined by
x
(Au)(x) = \ u(t)dt , 0 < x < 1 .
0
Consider the problem of finding u £ X from
(8.1.4) Au = f .
This problem is ill-posed because condition (iii) is not satisfied: in
j= o_ c c i ■, sin irnx
fact for f (x) = we have
/¥n
max I f I -» 0 as n -» «> ,
[0,1] n
max I u I = /ttH -» «> as n -» «> .
[0,1] n
Let us suppose that we try solutions of (8.1.4) that are bounded together
with their first derivative, that is
(8.1.5) max lul < 1 , max lu'l < 1 .
[0,1] " [0,1]
Denote by K the set of functions u £ X satisfying (8.1.5). K is by the
Ascoli-Arzela theorem a compact subset of X (see [Ta-La,1968; P- 295]).
A being a continuous operator condition (8.1.5) guarantees that the
inverse of Al„ is continuous. But by a direct calculation, we can find
' -1
an estimate of the modulus of continuity of (A|R)
By (8.1.4) we have u = f'. For every x £ [0,1] and every £ £ [0,1]
there exists an n between x and E, such that
fE) = f(x) + (?-x)f'(x) + f"(n) (x~ 5)
hence
u(x) = f(x) - fE) + U,(T1) UzSL ,
x - e; 2
|U(X)| < 2max .If I + lx_^X
" Ix- ?l 2
where max If| is over [0,1]. Now, supposing max If I < 1/16 and choosing
5 = x + 2 (max If I) 1/2 for x£ [0,^] ,
5 = x - 2 (max|f|I/2 for x£(|,l],
we obtain
|u(x) I < 2 (maxlfI) 1/2 .

158
Therefore
(8.1.6) || (Al)  f || < 2 || f II 1/2
(for small II fll ). We observe that to find this estimate,we only use
the second inequality of (8.1.5).
The choice of our particular values of E, is motivated as follows:
2F d
Put max|f| = F, |x~CI = d, and take d so as to maximize ^- + y ■
8.2 . Preliminary Discussion of the Stability of Abel's Equation
In this paragraph,we discuss in more detail the ill-posedness of
some physical problems leading to Abel equations. We refer to Chapters
2 and 3 for the descriptions of the problems discussed here. We shall
find a priori bounds on the solutions that have a clear physical
meaning and, as we shall prove in 8.3, guarantee the continuous dependence
of the solution on the data.
8.2.1. Abel's Mechanical Problem
As stated in Chapter 1 and as can be reformulated from 2.2 , Abel's
mechanical problem leads to the equation
x I 2
(8.2.1) J" XlJ±llL£i. d? = f (X), o<x<1 .
0 Vx-5
We recall that this equation formalizes the problem of finding a curve C
in the vertical plane that is the graph of an increasing function
x = ¢E) such that the falling time of a body (fixed to this curve and
f (x)
falling under the influence of nravity) is a known function —33 of the
\/2g
height x from which it falls.
We suppose $@) = 0. A suitable space where to Iook for a solution is
th<= space of functions with piecewise continuous derivative in [0,1].

159
^5
Fig. 8.2.1
(A) We inunedlately observe that the problem itself suggests a first
a priori bound of the solution: <j> should be an increasing function, that
is <J)'(x) > 0 for every x€ [0, 1 ] . without this itionotonlcity condition the
solution is not unique (if we allow isolated discontinuities of $').
In fact, for f(x) = \JSx every function <j> continuous and piecewise linear
of type x+c or -x+c solves (8.2.1). See Fig. 8.2.2 .
> x
Fig. 8.2.2

160
There may also be non-uniqueness of a continuously differentiable
solution. If
iaoth functions
f(x) = /(x-C)72 A+A-252)I72 d5
(x) = x(i-x) for 0<x<1 ,
,(x)
EcA -x) for 0 < x < 1/2
'j + (x--l) 2 for 1/2 < x < 1
are continuously differentiable in [0,1] and solve (8.2.1 ) .
(B) The problem of solving equation (8.2.1) is not well-posed even
if we suppose that <)>' > 0. In fact our next example will display a
sequence (g ) of admissible data such that
lim sup lg -gl = 0
n-«> [0,1]
where g is also admissible, but <|>n -f" <t> in the norms of L [0,1] and
L @,1). v«e even have
n l/[0,l] l'[o,i]
Here <j) and <$> are the solutions of (8.2.1) with f = gn or f = g,
respectively .
Example 8.2.1: Let h(x) = 2x, x£ [0,1] and let hn be defined as
k
r k , k 2k +1
x + - , for - < x < „
n n - 2n
hn(x) -\
, k+1 , 2k+1 k+1
3x - __, for -^- < x < -^
for k = 0,1,2,
See Fig. 8.2.3

161
^X
We have
hence
(8.2.2)
Furthermore
h' (x) = «J
Fig. 8.2.3
|h (x)-h(x)l < 1/2 n for 0<x<1 ,
lim suplh -h| = 0
n
n-K=°
c k 2k+i
1 for — < x < -
xi - n
-, c 2k+1 k + 1
3 for —=— < x <
2n - n

162
and
(8.2.3) 1 < h'(x) < 3 for x e [0,1].
Now we use a result which will later be proved as Theorem 8.3.3
Theorem: Let u iolve. the equation
1 r u (t) dt c , , _
— J -^37- =f(x) , 0 < x < 1 ,
\/n 0 \/x-t
and iuppoie f@) = 0. Then [with II II on, no Km In L @,1) me nave.
(8.2.4) nun < — (\\ f n 1/2+ n fii 1/2 Vn f w]]1 .
OO —
v^
Let g ,g be defined as solutions of
x g (t)dt
J ZZZ~ = V^fr h (x) , 0 < x < 1 ,
0 \/x-t
; cMj^t = ^-h(x) ^ 0<x<1 .
0 \/x-t
We have |h'-h'| = 1 and therefore by (8.2.2) and (8.2.4)
n
lim sup |g -gI = 0 .
n-«> [0,1] n
Here we have applied (8.2.4) with f = h -h and u = 9~9 •
Now let us consider the integral equations
x /l+<t>'2(?)
\ =^ d? = g (x) ,
0 Vx-5
; vA+^m d? = g(x)
o Vx-5
Their solutions are
£„(x) = J" \/h'2(T)-1 dx ,
0
<j>(x) = \/3 x .
This can be seen most simply by working formally with the operators J
1 /2
and D . We have
1/2

163
1/2 -
J17 g = v/nh, \A
nJ1/2 /l+<j>'2 = g,
A .,2 1 „1/2
\/TT
-L D1/2 (^ D1/2h)
V'n
Dh
Analogously for d> , g , h
3 J n 3n n
It follows that
(x)
"ky/2
n
- k 2k+1
for — < x < —~—
n - 2n
kv^2 js ( 2k + i\ , 2k + i k + 1
-1— + \/8 I x - —.—- for —~—- < x <
■ n \ 2n I 2n - n
Fig. 8.2.4
Posing § (x) = \fl x , we obtain
sup l<J>n-<(>l
\/2
2n
Therefore ji -»J in L -norm. But <(> *
0 as n -» oo.
it . Furthermore
II <t>nH , = / <j>(x)dx - ^| = J $(x)dx = II ?ll * II <HI .
1 0 0 '

164
The inverse of the operator in the left-hand side of (8.2.1) is
continuous in the L°°-norm at f with f (x) - 2 \fx (we leave this as an exercise).
We remark that the condition m < <j>' < M also does not lead to-a well-
posed problem.
(C) Since the natural a priori bound <)>' > 0 cannot restore the
stability in the solution of the problem (8.2.1), we need some alternative
a priori informations about the solution <)> . These could be of various
type. We only discuss some of them. One natural a priori information
could be that the slope of the graph of $ have not too big variations and
be neither too small nor too large. Let us assume (a) and (b) satisfied.
(a)
Idx
>" (x)
< M for x £ [0,1],
(b)
ml *
>" (x)
< m? for x £ [0,1].
These two conditions imply
(8.2.6)
dx
/l+<j>'2
< -^- for x £ [0,1] .
1
They have a clear physical meaning. In fact, if the slope ,,,, of the
graph of <j) rapidly changes, or is too big, it seems very unlikely that
the particle gliding on its profile could remain bound to the curve. On
the other hand, if the slope is too small the friction could stop the
body .
(D) Another type of conditions is based on convexity for the graph
of ¢, that is
(c) <)>" > 0 for x £ [0,1 ]
(d)
< 0 for x e [0,1].
We observe that the conditions (c) or (d) together with the following
condition
1 1
(f)
give
(8.2.7)
1
\
0
>' A) ' <t>" @)
A^P
(x)
> m > 0
dx <
-, I 2
2</1 + m
8.2.2. Inversion of Seismic Travel Times
For a first approach to the problem of inversion of seismic travel
times we refer to the plane model (see 5.2.3) . We recall that the unknown

165
in this problem is the velocity v of seismic waves in the earth's
interior. The data are the travel times and the exit points of the waves. The
model assumption is that the velocity of a seismic ray depends only on
the depth below the surface, of which it is an increasing function.
If we pose
1 1
(8.2.8) w = — , z(w) = inverse of w(z), w
the equation is
w
r WZ (w) •, 1 , -
J ——±Zz dw = 2 T (p)
p & -2
/w -p
where x depends on the ray parameter p. With the new variables
x = 1 - (-f/ , t = 1 - (-2-/
w
o
we obtain the classical Abel equation
x z (w vM-t)
(8.2.9) w f -—§-— dt = t(w vT^x) , 0 < x < 1
° 6 yS=t
Now by the hypothesis that v is increasing and in view of (8.2.8),we
obtain
(8.2.10) A z(Wq ^--£) > o .
Furthermore
(8.2.11) z(wQ s/l^t) |t=Q = 0 .
Now, recalling that in our model the depths considered are not too
large (in any case smaller than the diameter of the earth !) we can
suppose that
(8.2.12) 0 < z (wQ \/1-t) < E
where E is a known positive number,
8 . 2 . 3 . Other Examples and Instability Properties
For a discussion on the problem of spectroscopy measurements we refer
to 3.1 where we have stressed the usefulness of certain a priori bounds
on the solutions, which are similar to those in 8.2.1 and 8.2.2 .
To conclude this paragraph we assert that in many physical
problems (in the mechanical problem we pose u = y1+ <)>' 2 ) we can assume,on
the solution u of the Abel equation,some a priori bounds of the type

166
(i) |u(x) I, |u' (x) I < M (condition 8.2.6 for the mechanical problem).
(ii) c < u < c_ and u' with constant sign for the mechanical
problem, or the seismological problem , or in spectroscopic
measurements).
It is very simple to verify that if I is a compact real interval
the set of functions u fulfilling (i) is contained in a compact subset
of C (I). Furthermore the set of functions u fulfilling (ii) is contained
in a compact subset of L (I). In fact if u verifies (ii) we have
\ (|uI + Iu' I )dx < E
I
where E is a constant. The set
A = {u £ L1 (I) : \ (|u| + |u' I)dx < E}
I
1
is compact in L (I).
Now since the Abel operator J -- is continuous from L (I) to L (I) (see
Chapter 4) a theorem of general topology (see [Ti-Ar,1977] and 8.1) gives
t1/2
IA
u of
J1/2 u = f
depends continuously on f in the aforementioned function spaces. We shall
find explicit stability estimates in 8.3 .
Now we want to illustrate the instability of the Abel equation for
some a priori bounds different from (i) and (ii). Many of these are
similar to the extra informations we have in the case of spectroscopic
measurements. More precisely we prove that if the solution u of the equation
(8.2.13) _j x u(t)dt
satisfies only one of the following "extra conditions" there isn't
stability in L^-spaces. The extra conditions are the following ones:
(j) u(x) > 0 .
CD) C-, < u <x) < c2 •
(jjj) u' (x) > 0 or u' (x) < 0
(jv) u" (x) > 0 or u"(x) < 0 .
Example 8.2.2; For
f (x) = <n+1)' xn + a , 0<x<1, n> 1
n r(a+n+1) '
the solution u of the equation

167
J U = f
n n
is (use Euler's beta integral, see also Example 1.1.1]
We have
un(x) = (n+1)xn .
u >0, u'>0, u">0
n - n - n -
and for n -» «>
1-a-l
ii f j, = <n±UJ ^2
L @'1) r(a+n+D[p(n+a)+1]1/p
P
by Stirling's formula.
Furthermore (for all 1 < q < «>)
II u || = n , , .
n 1,9@,1) (nq+1I/cf
Now II f II -+0 for n -»oo if 1 < p < -^- , but II u II 7*0 for
n LP@,1) 1 a n Lq@,1)
n->«>. Therefore, none of the a priori bounds (j), (j j j) , (jv) yields
stability for the Abel equation (for u'< 0 or u" <0 we can consider
-fn and -un).
To show that condition (jj) cannot assure stability in Lp-spaces we
can use example 8.1.1 with a few small changes.
Example 8.2.3: Let
? a
f (x) = r(^ + li for 0 <x < 1 ,
f (X) = LJLT^aL sin nnx + f or 0 < x < 1 ,
C (nn)a " "
where
C = sup / 5° e 1K d5 .
[0,+oc) ' 0
Then the solution u of equation (8.2.13) is
u (x) = 2 for 0 < x < 1 .
The solutions u with f instead of f being
n n

168
, nnx
u = l ; cos(nnx-g) ds + 2
n C 0 ?°
we have
and
but
I 1 un(x) - 3
II f -f II -» 0 for n->«° ,
11 L~@,1)
lim II u -u||i > -^ .
n-x» L @, 1)
It follows that condition (jj) doesn't assure continuous dependence of
the solution of (8.2.13) on the data.
8.3. Stability Estimates for Solutions of Abel-type Integral Equations
8.3.1. Auxiliary Lemmas
In this paragraph we shall prove stability results for the general
linear Abel integral equation of first kind
(8.3.1) 1 ; *(*,t)u(t) dt = f(x), 0<x<1 ,
l0" 0 (x-t) ' a
supposing that in LP-norm A <p<°°) the first derivative (in subparagraph
8.3.2) or the second derivative (in subparagraph 8.3.3) of u is bounded.
Among other things,we prove Theorem 8.3.3 that we have already used in
Example 8.2.1. In subparagraph 8.3.4,we shall apply our stability
estimates to the case of discrete data. First we give some lemmas that will
be useful in the sequel.
Lemma 8.3.1: Le-t 0 < h < ■=■ and
. x+h
/8.3. 1) uh(x) = ^ \ u(t)dt ion 0<x<l-h . Tke.n tke. following two
e.&t£mate.& kotd.
(8.3.2) || u-u II < ^ llu'N
LP@,1-h) z LP@,1)
(8.3.3) || ull < 2h || u' II + || u||
LPA-h,1) LP@,1) LP@,1-h)
Proof; Young's inequality for convolutions (Theorem 4.1.2 with
g = u', f(x) = (*- +DXr0 hn (x) , q = 1, hence r = 1) gives (8.3.2).
Concerning (8.3.3) we only treat the case 1 < p < °°, the case p = °° being
analogous. Triangle and Young's inequality yield

169
1 _ 1/P 1 D 1/P
Hull < ( / |u(x)-u(x-h) |pdx) + ( \ |u(x-h)|pdx)
LpA-h,1) 1-h 1-h
1 x 11—h
.(/ (/ lu' (?) |d?)PdxI/p +(/(/ lu' (?) |d?)Pdx) 1/p
- 1-h 1-h 1-h x-h
1-h ,
+(/ |u(x)|Pdx)l/p
0
< 2h l| u' || + || ull
LP@,1) LP@,1-h)
Lemma 8.3.2: let uh be defined by (8.3.2). Vol O<0<1,1<p<+°°,
0<h<-2, we then have. the. following ebtlmatet,.
(8.3.5) || u-u || < hG|ulfl ,
n LP@,1-h) °'p
(8.3.6) || ull ^ < 2|u|., hG + || ull
LPA-h,1) U,P LP@,1-h)
Remark : For the definition of I u I . see 4.2.2.
0,p
Proof of (8.3.5). If x e [0,1-h] and -+-, = 1 then
P P 1
x+h 0—, x+h , . ._ . . |
lu(x)-uh(x)| <1 / |u(x)-u(t) Idt = h p / '"'q^/p "dt
x x h
.«»e^ "h 1°11%1¾1«<- -e<x* "■'•■^ia1"',/p ■
X Ix-tI r x Ix-tI r
the latter inequality < following from Holder's inequality. Therefore
„ 1-h x+h . . , ... ,p 1/p „
nu-uhn _<h0(/ (/ 'u<*>;u+ff'pdt)dx) <_h0iui0
Lp@,1-h) 0 x | x-tI ' p ,p
Proof of (8.3.6). By the triangle inequality, we have
1
II u|| < ( / |u(x)-u(x-h) |pdx) 1/P + II ull
LpA-h,1) 1-h LpA-h,1)
1 x
< ( / |u(x)- 1 / u(t)dt|p dxI/P +
1-h x-h

170
1 x ,
+ ( \ lu(x-h) -1 J" u(t)dtlp dxI/p + Null
1-h n x-h Lp@,1-h)
1 X
= ( / I c \ (u(x)-u(t))dt|p dxI/P
1-h1 x-h '
1-h . x-h |p ,
+ (/ N- / (u(x)-u(t))dtr dx)'/p + Null
1-2h' x-2h ' Lp@,1-h)
<2|ulnhG+||ull
°'P LP@,1-h)
Lemma 8.3.3; let he @,1/4] and f £ C°[0,1 ] With f' £ L°°@, 1) . Let u
iolve. ike. equation
(8.3.7) J1/2u(x) = -L ; u'^ = f(x)
\]v 0 \/x-t
w-c-Cfe f @) = 0 . Then the. following e.it-imate.6 kotd.
(8.3.8) || u-u II < _1 || f ■ || . hl/2
L°°@,1-h) \/n L @,1)
(8.3.9) II u || ^ < J_ || f ■ || ^ hl/2 + || U||
L°°A-h,1) \/tt L°°@,1) L°°@,1-h)
Remark: u, is the function defined in (8.3.2)
Proof: We have
. x+h
uh(x) -u(x) = ^ \ [u(t)-u(x)]dt for 0 < x < 1-h .
x
Since
,, 1 x f ■ (t)dt
u (x) = — J —r==-
\/tt 0 \/x-t
we have
^ / [u(t)-u(x)]dt = -f- / \/X+h-T • f ' (T)dT
V \/TTh *- V
+ / ( V^X+h-T - V/X^T ) f ' (T)dT } .
Denote the first of these integrals by I , the second by I_. Then
II. I < II f II • h3/2 .
1 " L~@,1)

171
Furthermore, by the concavity of the function t->\/t, we have
h
2v^f-T
(i/x+h-x - \/x-t) > 0 ,
(\/x+h-T - \/x-T ) <
2\/x+h-T
Therefore
I
I?l 5 / ( (\/x+h-T - \/X-T ) } If (T) IdT
0 L2\^? J
u x
1
1
)}
If (T) IdT
/X-T \/x+h-T
Using Young's inequality for convolutions,we obtain
H2I < II f II
3/2
L @,1)
Putting together these estimates we obtain (8.3.8).
To prove (8.3.9),we observe that
|u(x) | < |u(x)-u(x-h) I + || u|| _
L @,1-h)
Now
1
u (x) -u (x-h) = — \
f (T)dT
\/tt x-h \/x-t
x-h
/TT n ^ \/v —T \/v—Vl — T-1
(T)dT .
\/ti 0 L \/x-T \/x-h-TJ
Ubiny the same arguments as for !_, we obtain
lu(x)-u(x-h)| <
4|| f II
L°°@,1) , h1/2
This proves (8.3.9).
8.3.2. L^-bounded First Derivative of the Solution
Theorem 8.3.1: I <5 a £ @, 1) , pE[1,+»], u £ LF@,1) , u' £ LF@,1 ]
and
(8.3.10)
we. have.
J U = f

172
(8.3.11) II u||p < C1 (a) ill u' ||p1+a + II f II p 1+a | II f||p1+a .
I<5 ^utitkenmotie u £ W ,p@,1) (, ox a value. 0£ @,1) and a value p £ [1,+°°)
then
(8.3.12) || u||p < C2(a,0) {|u|0G;pa + || f||p0 + a } || f llp0 + a .
The comtanti C (a) and c2(a,0) atie computable and depend on the Indicated
arguments, only.
II f II
TjC c j_i> iu^lclently bmall, (8.3.11) can be blmpllhled to
llu'llp
(8.3.11 ') II u|l < C, (a) II u' II 1+a II f M 1+a
a _1
-,,-, ,, - ,, ,+a II fll 1
1 ' "p p
q n
Concerning W ,Fand I -l^ see 4.2.2 .
0,P —
Proof: Equation J u = f is equivalent to Ju = J f (remember:
x
Ju (x) = \ u(t)dt for 0 < x < 1) . Now for 0 < h < 1/2 and
0
. x+h
(8.3.13) uv,<x) =i I u(t)dt, 0<x<1-h ,
h
x
we obtain
(8.3.14) uh(x) = 1 |j1 a f (x+h)-J1 a f(x)| ,
Ml-a) 1 J
whence
x+h
uh<x) = hr;,.., -i \ (x+h-t) a f (t)dt
; | (x+h-t) a -<x-t) a| f(t)dtj
Young's inequality for convolutions(Theorem 4.1.2 and [Ha-Li-Po,1978] '
gives
5, -a
(8.3.15) II u II < -^ II f||
Lp@,1-h) TB-a) Lp@,1)
Now, by the triangle inequality, and lemma 8.3.1 we have
II u II < II u|| + II u||
Lp@,1) Lp@,1-h) LpA-h,1)

173
< 2|| u || + 2h II u * II
LP@,1 -h) LP@,1)
< 2|| u -u II + 2llu II + 2h||u' II
LP@,1-h) nLP@,1-h) LP@,1)
< 5h II u * II + 2II u || ,
LP@,1) n LP@,1 -h)
Using (8.3.15) we obtain
Hull < ±5 || f || + 3h llu' II
LP @,1) rB-a) LP@,1) LP@,1)
By minimizing the right-hand side of this inequality for he [0,1/2] we
obtain the estimate (8.3.11). The estimate (8.3.12) can be found in an
analogous way using Lemma 8.3.2 .
Remark 8.3.1 : ["'e shall show by an example that the exponent 1/A+a)
is the best possible HSlder exponent for II f || in the stability estimate
(8.3.11) . P
Let K = {u e LP@,1) : llu'll = 1}
LP@,1)
where 1 < p < +<». Clearly inequality (8.3.11) is equivalent to
a 1
llu ||p < C1 (a) {1 + IHau ||p1 +a } IUau|lp1 +a
for u e TK . Now we prove that there exists a sequence (u ) in K such
that
(i) II Jaunllp - 0 ,
"Vp
(ii) iim ,fi/(i + otr = const > ° •
n-><» 11 J u 11
n p
We consider only the case 1 < p < + °° , leaving to the reader the
treatment of p = + ». Take
(x) . [P(n-1)+1]1/pxn for n=1,2,3,... .
n
Then all u e K and
n
1/P
j«u (x) = £' [p(n-D +1] xn + a
n r (a) nT (a + n + 1 )
We obtain
Mu i, = i r p(n-nti t1/p n. 1
n p n pn + 1

174
IUau II
n!
n p nr (a)T (a +n + 1;
n!
p(n - 1)+ 1
because, by Stirling's formula,
p(n + a)+ 1
1
1/P
1
T (a)n
1 + a
r (a + n + 1 ) n
From these asymptotic relationships,we get
lim —
n-w IIJ "u
n -E.
1
1/A +a)
(Ha))
1 + a
n p
The importance of Theorem 8.3.1 consists in restoring the stability
in many physical problems formulated as an Abel equation with an a priori
bound (see 8.2)
i|u,|IlP(o,d 2 E
(8.3.16)
for some p £ [ 1 , +«>] .
By the a priori bound and the estimate (8.3.11) the stability is
restored. Let us consider for example the mechanical problem with the
conditions (a) and (b) (see 8.2.1 - C). We have
J. /1 +r2
dx
M
Now if we put u. = v1 + <ji'. , ¢. being a solution corresponding to the
data f., i = 1,2, then by estimate C.3.11) we have
M1/3 1/3
2/3
u.-u-ll < C { -—^=- + ||f - f || ' }||f-f2lloo
1 Z L°° @,1) " m//3 1 Z L°°@,1) L @,1)
Now by condition (b) of 8.2.1, we have
i i m9
II ¢, - ¢,, II < -- II u. -u-ll
1 Z L°°@,1) - ml 1 l L~@,1
Therefore, since ¢..@) - ¢,@) = 0 ,
II * -4. II „ + II *\ -^ H „
1 L @,1) l Z L @,1)
M
1/3
1/3
2/3
2m9
5^-1^73 * "'l - '2 "t. ,„,„1 ' -'»',-„,„
Theorem 8.3.2: Le.£ u £ LF@,1) , u' £ LF@,1) and
x
. u(xK-= —■—
a
(8.3.17)
A u(x):=a_L_ / K(x,t)u(t) dt = f(x) f x £ [QJ]
r (a) 0 (x - t) ! a

175
wke/ie a £ @,1], p £ [1,+ =] , and i, atli, {>lei>
(i) k, H e c°(t), wfee^e t = {(x,t) em2:0<t<x<l} ,
d t _ _ _
(ii) k(x,x) = 1 ion. x e [0,1 ] .
Then
a
a
1+a
1
1+a
(8.3.1:
II u II.
< C1 (a,p,M){ II u'll + llf lip } Nf I
wkene C. (a,p,M) -ii a compa-tab£e corci-tarc-t that depends on a, p and
M = sup
3K
only.
,T6'P
1M u -ii known to lie In w ,p@,1) ^o-t a value 6 £ @,1) and a ua£ae
p £ [ 1 ,+ °° ), then
a
6+a
a
6 + a
e+a
|p < C2(a,p,6,M){|ul6^p + llfllp } llfllp
(8.3.19) II u
wke/ie C„(a,p,6,M) li> a computable constant depending on the Indicated
an.a,umentt, only.
Proof: We observe that with I as identity in Lp@,1) and B define
by
Bav(x) = _2±!Lje° ; {v(?) ; — fH(x,t)-,_^ dt „ > ^5
r wu) / ^ "^--i -^-.
D ? 3t (x - t) ' a/ (t - ?r
where H(x,t) = K(x,t) -K(x,x),the operator
(8.3.20)
(I - B ) J
a
is from Lp@,1) to Lp@,1). For every v £ Lp@,1) we have
(8.3.21) N v II < C(M,p) || (I -Ba)vll
C(M,p) =-
X C (p) BM)
n=0
exp BM)
for 1 < p < +c
for p = + <*>
(see proof of Theorem 5.1.3 and Lenuna 5.1.1) where C (p) =p n/p(n!) /p
By (8.3.20) and (8.3.21) we have
IUau|l < C(M,p) || Aqu II
and the estimate (8.3.11) says that
(8.3.19) Mull < C1 (a){ II u* l|1+a+ II Jaul|1+a} IUauM1+a .
Combining these two estimates we obtain (8.3.18).
In an analogous way we can find (8.3.19).

176
Remark 8.3.2: If me Heptane the hypotheili (i) of Theorem 8.3.2 by
(l) K, || e C°(T)
the estimate* (8.3.18) and (8.3.19) atie a till valid but with constant*
C, and C, depending on a,'6,p and sup l-s— j (see [Ve,1983]).
1 2 ' D ' 'r rpt" I gxl
Now we present a stability result for the original Abel equation
(8.3.22)
jV2 u(x):=_i_ f Hiiidt = f(x) f o < x < 1 ,
/~tT 6 /x - t
that we have used in Example 8.2.1.
Theorem 8.3.3: let u iolve the equation (8.3.22) and iuppo£t that
f(O) = 0 . Then
(8.3.23)
II ull
< — I II f II
1/2
L @,1) /F - L @,1
+ l|f||1/2 )||f H1/2
I L"@,1)' L @,1]
Proof: For u, as defined by (8.3.2) we get, using (8.3.22),
h
rX+h
(x) ._!_ (xf iltidt + ? rj___ _ jl_\ f(t)dtl
h/ir l x /x+h-t 0 v/x+h-t /x-t7 J
Therefore by Young's inequality
lluv.ll
4l|f|lL~@,1)
h t°° //-. -i v.\ ~ a-,1/2
L @,1-h) /tt h
and by the triangle inequality and Lemma 8.3.3
Mull < II u -u, II + llivll
L°° @,1) " h l" @,1-h) ^ L~ @,1-h)
+ II ull
32 h
L A-h,1)
1/2
llf'll +
L°° @,1)
8l|fllL"@,1]
/TT h
1/2
1
Minimizing the right-hand side with respect to h £ [0,-j] we get (8.3.23)
8 .3 . 3 . IjP-Dounded Second Derivative of the Solution
Theorem 8.3.4: I (J a £ [0,1], p £ [1,+ =°] , u £ LP@,1), u" £ Lp@,1)
and
J u = f
then
(8.3..24)
a
2+a
a
a+2
lullp < C(a) jllu1'!! + II fllp ]H flp
2
a+2

177
wheie C(a) -Lit a computable, constant that depends on a only.
Proof: Let h £ @,1/6] and
1 x+h . /X+2h x+h \
(8.3.25) u (x) = ± / u(t)dt ~ 4r{ I u(t)dt - / u(t)dt),
x x+h x '
0 < x < 1 - 2h.
By the triangle inequality and Lemma 8.3.4 we obtain
II ull < II u - u II + llu II + II ull
LP@,1) n L'P@,1-2h) nLP@,1-2h) LPA,1-2h)
< 23hZ llu"ll + — || f ||
Lp@,1) TB-a) Lp@,1)
By minimizing the last expression with respect to h £ @,1/6] we obtain
the estimate (8.3.24).
Lemma 8.3.4: Le.t u and f be ai, -in Tfieo-tem 8.3.4. lit ube defined by
(8.3.25). Then the following eitlmatti hold.
7V,2
(8.3.26) II u -u, II < -^- II u"ll
n Lp@,1 - 2h) " b LP@,1)
(8.3.27) llu, II n < - h"a II f II
n Lp @,1-2h) rB - a) Lp @,1)
(8.3.28) Hull < 20h2 l|u"ll + II ull
LPA-2h,1) LP @,1) LP @,1-2h)
+ _ih___ ||f ||
r B -a) Lp @,1) .
x
Proof: For U(x) = f u(t) dt, we have, by Taylor's formula,
6
,2 1 x+h y
U(x + h) =U(x) + U' (x)h + U" (x) ^r + i f (x + h - t) U'" (t) dt
2 6 x
that
and
(8.3.
is
, x+h
E '
X
.29)
1 \ u(t)dt = u(x) + § u'(x) + —~ j" (x+h - tJ u" (t) dt
^ 2 bh
r i /X+2h x+h \-|
u (x) -u(x) = § ^u'(x) + -~ \ u(t)dt - f u(t)dt
n z l h Nx+h x /J
x+h -
4-/ (x + h - t)z u" (t)dt.

178
Now
1 /X+2h x+h \ . x+h t+h x
u' (x) ~-~ ( \ u(t)dt - / u(t)dtj = -~ / dt / dx \ u" (?)d?
x+h
/
x
x+h t+h
• J
h*1 x t x
Therefore
(8.3.30)
Furthermore
(8.3.31)
,x+2h
x+h
u' (X)
r u(t)dt - r u(t)dt
h '-x+h
/J|lL-p@,1-
2h)
< h'll u"ll
Lp@,1-2h)
1 x+h ? I! h2
4- / (x + h-tr u" (tjdt <-rllu"Mn
6h x !,LP @,1-2h) b LP@,1)
(8.3.30), (8.3.31) and (8.3.29) give (8.3.26).
The proof of the estimate (8.3.27) is analogous to that of estiirate for
II u, II in the proof of Theorem 8.5.1 (formula (8.3.15)).
h Lp @,1-h)
In order to prove (8.3.28) we observe that for 0 < h < 1/6 and
1 - 2h < x < 1 we have
(8.3.32)
2h
^x-2h
I 1 / x *~m \I
lu(x) I < u(x) - u(x- 2h) -— / u(t)dt - \ u(t)dt
" I ^n '-x-2h x-4h /!
I / x x-2h \|
+ j— ( \ u(t)dt - \ u(t)dt) + !u(x- 2h) I .
From
1 , x x-2h v . x t t
u(x) -u(x-2h) -4- \ u(t)dt - \ u(t)dt =— J dt J d? /u"(n)dn
m Vx-2h x-4h ' ^x-2h t-2h 5
we obtain
(8.3.33)
u (x) - u (x - 2h)
2h
x x-2h
f u(t)dt- f u(t)dt
x-2h
x-4h
]LF @,1-2h)
Furthermore
(8.3.34)
< 20h^ II u"ll
. a x-2h
~ \ u(t)dt - / u(t)dt
2h
x-2h
x-4h
LF@,1]
LF@,1-2h)
A 1 ^
_ih l| f ||
r<2-a) LP(o,i:
By (8.3.32), (8.3.33), (8.3.34) we finally get (8.3.28).

179
8.3.4.Discrete Data
In this subparagraph,we briefly consider the following type of
problem:
Let x^ = 0 < x. < . . . •£ x < 1 = x , , f. £ IR, 1=0, ...n+1,
0 1 n n+1' l '
and let e and E be given positive numbers. We want to estimate the
diameter of the set
H = {u £ LP@,1) : iJ^ufx.) - f, I < e , || u|| . ^ < E }
P'k lip- Wk'p@,1) "
where p £ [1, + «>] , I | is the discretized Lp norm; that is
n .
lip I = ( I lip.lp(x . - x )) /p, E is a positive number, k £ IR .
p i=o 1
We consider in detail only the case p=1, k £ A -a,1). As we can
observe by studying the proof of the next theorem, the estimate of diam
H , can be found by a combined use of smoothing estimates (Chapter 4),
stability estimates and simple methods of numerical integration. This
method is very general, but we don't apply it to every possible case.
Theorem 8.3.5: I (J k £ A - a , 1 ) than
—— a k
(8.3.35) diam H., < 2c2 (a ,k) c (a ,k) JEk+a +■ (EA + e) k+aj (EA + e) a+k
wke.ne. c_(a,k) li, the. constant o & <Li,ti.maX<L (8.3.4), c(a,k) li, the.
constant o<) mtlmaXl D.2.12) In wh-ick Q = k and A = max (x. i-xi'-
Proof: Let u. ,u2 £ H. , and w = u. -u?. Then
n
(8.3.36) 21 [Jaw(x ) | (x. - x ) < 2e, 1=0,1,...n+1 ,
1=0
(8.3.37) II wll < 2E.
W ' @,1)
With J w = g we have, by (8.3.37) and Theorem 4.2.3,
(8.3.38) II g' II < 2c(a,6)E
L1@,1)
where c(a,6) is the constant of inequality D.2.12). Now, posing
A = max{x. 1 - x.}, we have
1 n xi+1
II g II ., = / |g(x) Idx < Z / |g(x) - g(x.) I dx + 2e
L @,1) 0 i=0 x. x
x 1
n i+1 x
< T / I [ I g' (t)Idt)dx + 2e
1=0 x. x.

180
n xi+1 xi+1
= X / lg' (t) I / dx dt+ 2e < A llg' II . + 2e
i=0 x. t L @,1)
Therefore
(8.3.39) II gll . < 2(c(a,k) EA + e) .
L @,1)
Now, in view of(8.3.12) we find
a a
II wll < c,(a,k) { BE)k+a + [2(c(a,K)EA + e) ]k+a}
L'@,1) ^
k
' {2(c(a,k)EA + e)} k+a ,
which is the estimate (8.3.35).
Corollary 8.3.1: Von. the pan.tic.ulan. ca&e k = p=1 we have. the. e.i -
timate. ,
a a.„ t
{8.3.40) diam H < 2&1(a)c(a){E1+a+ (EA +eI+a}(EA + eI+a
whe.ne. C. (a) li, the. constant o & Inequality (8.3.11) and C(a) li, the. con-
it ant C(a,k) o i e.i> timate. D.2.12) $01 k = =j- .
Proof: With the same notation as in the proof of Theorem 8.3.5 we
observe that we have the estimate
(8.3.41) II wll , < -— II wll
w-''@,1) kA-k) w'' @,1)
In fact (by Theorem 4.1.1)
,w, = 2] (J '"(*>-w(t)l_dt) dx
k'' 0 0 lx-t|1+k
< 2 \ dX J" -^ T-r- J" |U' (?) IdS
0 0 (x- t)'+K t
1 x 5 ,
= 2 \ dx J lu' E) Id? /
0 0 0 (x-tI+k
{ } dx J lu'(g)l dg + } dx X |u,(?)|d?}
1 0 0 (x - 5) 0x0
< _J— II u. II
kA - k) L @,1)
By (8.3.40) we have (take k = ~-^ in (8.3.39) and (8.3.41!
II gll 1 < 2 (c (a)EA + e) .
l'@,1)

181
Therefore, by (8.3.11), we obtain (8.3.40).
Remark: To undenstand a possible usefaul application ofa the estimate ofa
diam H, , we should conslden the pnactlcal and numenlcal pnoblems that
K , p
ane nelated to Abel'is equation. In faact, In pn.actls.al situations, the
data f ofa the. pnoblem J u(x) = f(x) can be measuned only at a finite
set 0E points x.,x5,...,x , and they ane contaminated by an ennon ofa
maximal value, e. Wow, numerical methods [see Chapten 9) a££ow us to
fclnd pan.tlcul.an solutions ofa the pnoblem |Jau(x.)-f.| < e, but, as me
have shown In. many examples (see examples 8.1.1, 8.2.2), It Is not
possible to evaluate the distance oft the numenlcal solution to the exact
[on anothen) solution without an appn.opn.late a pnlonl bound on the
Solution. The estimates ofa type C.3.35) and (8.3.40) penmlt to estimate the
distance between a pantlculan solution and the exact {on anothen)
solution In tenms ofa: ennon on the data, an a pnlonl bound, and the dlstnl-
butlon ofa the. points x ,...,x

Chapter 9 • On Numerical Treatment of
First Kind Abel Integral
Equations
9.1. General Considerations
We briefly describe some methods for treating first kind Abel
integral equations and give a report on a numerical experiment. When choosing
a numerical method,one should consider the points (a), (b) and (c) which
are intimately connected with the origin of the problem.
(a) the type, ofj equation to be tn.ea.ted,
(b) the istKuctu/ie and pneclitlon oft the data,
(c) the coni>tn.aLnth the iolut-lon should Aatlifiy.
The nicest Situation is the purely mathematical problem involving an operator
equation Au = f,where the function f is given by a formula and the
function u is unknown. In this case, one may use as an approximation for u any
problem formulation equivalent to Au = f and discretize the most
appropriate one amenable to a particular numerical method. The attainable
accuracy of the approximation then depends on the smoothness of f
(provided f is in the range of A).
In real life applications, however, there usually is a
most natural form of the equation, and intuition advises ones to directly
discretize this form,taking into account the natural structure of the
data .
The most important forms of Abel integral equations are the
following ones (we omit trivial modifications) where a is a positive real
number or «> and 0 < a < 1
1 x -1
(9.1.1) yrj-y \ (x-t)a u(t)dt = f(x), 0<x<a.
K ' 0
1 X a-1
(9.1.2) -p4—r \ K(x,t)(x-t)a u(t)dt = f(x), 0<x<a .
K ' 0
1 x -1
(9.1.3) yrj-y \ K(x,t,u(t) ) (x-t)a dt = f(x), 0<x<a .
K ' 0
a ? -1/2
(9.1.4) / (t -y. ) u(t)dt = f(x), 0<x<a .

183
a -1/2
(9.1.5) 2 \ (t -x^) t u(t)dt = f(x), 0<x<a.
x
Let us comment en these various forms. The most important case in applications
is the one corresponding to a = 1/2, as in Abel's mechanical problem (see [Ad,1823]).
By obvious substitutions, (9.1.4) and (9.1.5) can be transformed into equations of
type (9.1.1) vvith a = 1/2. Other values of a arise in Sjjecial prcolems for Tricomi's
partial differential equation (see, e.g., [Ge-Wo,1986], [Bi,1964]). Equation (9.1.1)
with a = 1/2 occurs in several problems
of determination of potentials (see [Ke,1976]). For an application of
(9.1.2) with a = 1/2 see Anderssen et al. [An-Ho-We,1973].Equation
(9.1.5) is used in the spectroscopy of cylindrical gas discharges.
Equation (9.1.4) occurs in the Herglotz-Wiechert spherical earth model
of seismic travel time inversion (see [He,1907] and [Wi-Zoe,1907]), in
optical fibres [Ma,1979] and in spherical stereology (see [Wi,1925] and
[Re,1955]).
To the possible objection that the forms (9.1.4) and (9.1.5) are not
really different (in (9.1.5),one might consider 2t u(t) as the unknown
function) the answer is that in applications the behaviour of the
solution u at the origin may be important.
Although for (9.1.1) , (9.1.4) and (9.1.5) inversion formulas are
available numerical methods are needed either because the integrals
involved do not exist as elementary functions or because the function f
is only approximately given as a finite set of measured values. It is
also worthwhile to take these equations as model equations for more
general situations. In applications, there are often given only approximate
values of a finite set of linear functionals of f, in the simplest case
values of f at a discrete set of points. When they are given by a
measuring instrument,one appreciates if these points are equidistant. Thus
standard methods of (equidistant) discretization are called for, these
methods being particularly good and rigorously analyzable for their error
if the data function f is assumed (very) smooth. There are such methods
for any arbitrary order of accuracy.
Regularization methods are required if the data are (seriously)
contaminated by noise as,e.g. , in the case of spectroscopic measurements
or in stereology. If smoothness of the approximate solution is the
dominant requirement,one may use Tikhonov-like regularization schemes
(see e.g. [Ge-wo,1986 ] ) .However, as hinted at Chapter 3 qualitative
information like nonnegativity or monotonicity or convexity is often more
important, and such extra information should be incorporated into a
numerical approximation scheme.

184
A particular problem is the approximation of nonsmooth solutions
(or even distributions) which,in some instances ,better model the real
solution than smooth ones ([Go,1986 and 1987]).
9.2. Quadrature Methods
We shall sketch some methods for equations of type (9.1.2) under
the assumptions that there is a nonnegative integer m such that
K € Cm+1 {(x) | 0<t<x<a , K(x,x) = 1 for 0<x<a}
and Daf £ Cm [0,a].
Under these assumptions (9.1.2) has a unique continuous solution
u£Cm[0,a]. See Theorem 5.1.4 .
Physicists have for a long tine applied simple inter-
polatory quadrature methods which lead to triangular systems of linear
equations. There exists a vast literature on this subject,
in particular for equations of type (9.1.5). For such methods applied to
equations (9.1.2) a rigorous theory of convergence has recently been
worked out by P.P.B. Eggermont 1979,1981 and R.F. Cameron and S. McKee,
1985 .
We give a brief description of the midpoint product integration
method and of the trapezoidal product integration method, assuming the
upper limit a in A.2) as finite.
Let
h = a/N, x± = ih, f± = f(x±) .
For the midpoint method replace the product
K(x,t)u(t) by K(x,t)u(t)
where
K(x,t) = K(x,x .), u(t) = u . for x. .<t<x. .
Then collocate (9.1.2) at the points x. by the formula
x.
1 1 -1
jj^y \ K(xi,t)u(t) (x^tH ' dt = f. for i=1,2,...,N.
With K .= K(x.,x ) we arrive at the triangular linear system
i,J+2 1 3+2
, a i-1 / n
r(a + i) Z K 1 (i_j) ~ ^-3-1)° u 1 = fi ' i = 1,2,...,N ,
j=0 i, j +^x J + 2

185
for the determination of the unknown values u for j = 0, 1 , 2 , . . . ,N-1
3 +1
One hopes that |u - ulx U is small if h is small.
3 +1 K 3 +I
More precisely we have the following convergence result [Eg, 1979].
I<5 the iiOlut-ion Of) (9.1.2) hai, a LZpAchA.tz contA.nu.oua> de.tiA.vatA.ve.
A.n [0,a] convergence o<5 the approximate Aolu.tA.on to the exact solution
A.i> 0E onden. h , mole ptiecA.Aely-. thene It, a constant M Mich that
lu - u(x ,11 < Mh1+a ion. j = 0,1,...,N-1 .
3+? 3 + 2
For the trapezoidal method replace, for x=x.,x.<t<x. .,
r ^ ' 1J--3+1
the expression K(x.,t)u(t) by a linear interpolation ansatz
(x. ,.-t)K. .u . + (t-x.)K. . . u . . /h
:+1 1,: : :' 1,:+1 :+1/ '
where K. . = K(x.,x.) and collocate (9.1.2) at the points x. for
1,:1: 1
i = 1,2,...,N . Again a triangular system of linear equations results for
the unknown values u. which hopefully approximate the values u (x .) .
Eggermont,1981, has shown that the differences |u(x.)-u.| are uni-
2 3 3
formly 0(h ) if the kernel K and the solution u of (9.1.2) have Lipschitz
continuous second derivatives.
Arbitrarily1 high order of convergence (measured as a power of h) in
the discrete maximum norm can be obtained by fractional linear multistep
methods. These are based on discrete multistep formulas for approximation
of the fractional integral operator, and their theory has been worked out
mainly by Lubich,1987.( See also the comprehensive book of Brunner and
van der Houwen,1986) . There is a strong temptation to go into this fas-,
cinating theory, but we must resist it and refer the reader,to the bibliography.
Branca,1976,analyzes a piecewise linear interpolation method for the
nonlinear equation (9.1.3) in the special case a = 1/2 and shows it to be
2
0(h (-convergent in the discrete maximum norm under appropriate
smoothness assumptions. By a piecewise quadratic interpolation technique,he
3
obtains an 0(h (-convergent method for this equation.
9.3. Evaluation of Measurements
The methods of the preceding paragraph, in particular the ones of
high order accuracy, are very useful if f or u is srauyth enough ana f is known very

186
accurately (e.g. given by an explicit formula). If, however, f is only
incompletely or inaccurately given by measurements, there is unavoidable
noise. This noise is amplified by the methods of 9.2, as the
computational grid is taken finer and finer. Even worse, shape properties (for
example non-negativity) which for physical reasons the exact solution has are
often not reproduced by numerical calculations. Some kind of regular-
ization is required.
To be sufficiently general,we write any of the linear integral
equations (9.1.1), (9.1.2), (9.1.4), (9.1.5) in the form
(9.3.1) Au = f
with an appropriate linear operator A.
As a model of the measuring device, let us assume linear functionals
a. for j = 1,2,...,J acting on f but perturbed by unknown random errors
p. . As available information, we have the values
(9.3.2) aj = <oyf> + Pj, j = 1,2,...,J .
In the simplest case we have point evalutions
(9.3.3) <a./f> = f(x.), 0<x.<x9<...<x.<a,
In a more realistic model, functionals of the form
a
(9.3.4) <a,,f> = { w.(x)f(x)dx with wi > 0
are to be considered.
We want the approximate solution u to share one or more of the
important properties the exact solution u is known to have,e.g. smoothness,
non-negativity, monotonicity, convexity, unimodality. With linearly
independent functions u for n = 1,2,...,N and functions f = A un(because
A is injective these are linearly independent as well) take
(9.3.4) u = I c u^ ,
, n n '
n=1
with the coefficients c to be determined. Correspondingly
N
(9.3.5) ? = I c f
n n
n=1
We wish, of course, u«u, ? sa f .
Our problem can now be formulated as one of optA.mA.zatA.on-. to &A.t ?
to ike. data In Auck a way that the. extna condA.tA.oni u AatA.A^A.e.A ane.

187
I approximately) ^ul^llled {,on u, the fitting being achieved by minimizing
an appn.opn.late mea&une 0(J deviation.
The unconstrained GauAA Leakt AquaneA fill consists in minimizing
the quadratic form
(9.3.6) Q(c) = I y (a -
J
I
j-1
<a.,f > ) , c tffi
where the positive weights y. are prescribed (all = 1 in the simplest
-* J
case). Here c denotes the column vector with components c.,c_,...,c„.
The integer number N may be smaller or larger than or equal to J, the
particular case N = J meaning interpolation (Q(c) = 0) .
The quadratic function Q(c) has a unique minimizer c if and only if
its second degree part is strictly positive definite. To find a necessary
and sufficient condition for positive-definiteness we write Q(c) in matrix
vector notation with
/
\ aj/
as data vector, c
as vector of
coefficients,
<a1,f.> <a1,f?>
<a2,f.> <a?,f?>
«VfN>
<a2,fN>
^J'V <aj'f2>
<ajrfN>
as Gram-matrix, and the diagonal matrices r = diag(y.,y2,
as weight matrix,
/Y-
,1/2 _
liag(\Zy1 , \ff^,
> ^J>
We denote transposes of vectors and matrices by the superscript T. Note
that all quantities are assumed to be real.
With these notations a straightforward calculation using (9.3.5)
and changes in orders of summation, yields
(9.3.7) Q(c) = (T1/2 M c)T (T1/2 M c)
(Ta)T M c + (Ta)T a
The second-degree term
(r1/2 Mc-)T<r1/2 mS)

188
is nositive definite if and only if the matrix M has rank N.
We also see that this necessary and sufficient condition can be met
only if J>N, i.e. if there are no less data values then there are
coefficients to be determined. And furthermore we see that if the a .'s are point
evaluation functionals (<a,,f> = f(x .)) then for equations (9.1.1),
(9.1.2) none of the x. should be equal to 0, whereas for (9.1.4) and
(9.1.5) none of them should be equal to a. Otherwise for the corresponding
index j we would have f (x.) = (A u )(x.) = 0 for n = 1,2,...,N, hence
a line of zeroes in the matrix M.
To apply this method a good choice of basis functions u or f is
essential. It should be made in such a way as to yield a Gram matrix M
of rank N. Gorenflo and Kovetz, 1966,and Minerbo and Levy, 1969,take for
the spectroscopy equation (9.1.5) polynomials multiplied by a suitable
function and N small, thus achieving "smoothness" (in a colloquial sense:
oscillations suppressed). Disadvantages: low accuracy, choice of larger
values N leads to unwanted oscillations. In the problem treated by
Gorenflo and Kovetz the exact solution u was known to be everywhere
non-negative. By forcing u(x) >0 at a (large) finite set of points (this leads
to a quadratic optimization problem) they obtain u "approximately" non-
negative.
Today it is well-known that piecewise polynomials ("splines" in the
general sense of the word) are better. We recommend a very robust
kind of approximation, namely approximation of u by continuous piece-
wise linear ansatz. This means taking the functions u as hat functions
(or roof functions):
With
0 = t- < t, < t-
< tN = a
take
(9.3.8)
u (t)
n
u (t)
n
u (t)
n
0 for
t-t ,
n-1
t -t .
n n-1
t .-t
n+1
t .-t
n+1 n
t < t
n-1
and t > t
n+1
for
for
t . < t < t
n-1 -- n
t < t < t .
n - - n+1
if n e {2,3,...,N-1 } and omit in these definitions everything outside
[0,a] in cases n = 1 and n = N
0,1,
k * n .
[t.,tj+1], j
Every u is linear in each subinterval
J n
, N-1, and all u (tn) = 1, whereas un(tk> = 0 for

189
0=t
Fig. 9-3.1
Because of measurement errors ,we cannot expect a high order of accuracy
for the approximate solution. So, why should we use higher degree splines ?
Numerical experience shows that these splines of degree 1 are good enough.
However, the main advantage-, of using the hat functions as basis functions
u is the ease they offer in incorporating customary shape conditions as
constraints in form of linear inequalities for the coefficientsc which
now coincide with the values u(t
We now have :
N0ny1e.gat-Lvj.ty 0(J u
N
I
n=1
c u
n n
-ii equivalent to
(9.3.9) cn - ° ^0K n = 1'2'---'N
Monoton-ic -lnc>ieat>e o<5 u .ii equivalent
to
(9.3.10)
'n+1
c > 0
n -
\ofi n
1,2,...,N-1
Convexity 0& u Lis equivalent to
(9.3.11;
c _-c
n+2 n+1
xn+2-xn+1
c , -c
n+1 n
x , -x
n+1 n
> 0 ioi n = 1,2, . . . ,N-2
The use of such inequality constraints has some regularizing effect
as may be seen in 9.4. Another possibility is a discrete Tikhonov regula-
rization. It consists in minimizing the quadratic function (we describe
it for hat functions as basis functions u )

190
N-1
(9.3.12) Qx(c) = Q(c) + X X
n=1
(with or without additional inequality constraints) with a suitable
poi-it-ive. parameter X. For any X>0 the function Qi(c) is positive
definite.
Still another method is the "tizguZcuUzcutlon by diicmtizcuLLon". This
means that the coarseness of discretization itself has a regularizing
effect, that is, there is an optimal value of N which in this case
acts as a kind of regularization parameter. Compare Natterer, 1977.
In 9.4 we shall present results of a numerical case study in which
these basis functions have been used.
We conclude this paragraph by giving a short account of two
interesting variants for treating the spectroscopic equation (9.1.5) in case
of known non-negativity of the exact solution. These variants have been
proposed by F.M. Larkin, 1969 and,by numerical tests,shown to work well.
The first variant is as follows:
In an appropriate function space U (representing the set of
candidates for the exact solution) he determines an element by a maximum
Z-Lke.Z£kood m>£A.ma£A.on technique. In his study the values <a,,f>
are of the form
x . .
\ f(x)dx where 0 = x <x. < ... < x„ - a .
x ,
:
After deriving conditions for the measured values to be compatible with
a non-negative solution u (if they are not they must be modified by a
special preprocessing technique and replaced by compatible values, the
modified data vector being as near as possible to the original one), he
arrives at the problem of minimizing
a
J" u(t) log u(t) t dt under the restriction that u should
0
produce the values a. (or the modified ones, respectively).
~ 2
As a second method,Larkin proposes an ansatz u(t) = (h(t)) so that
definitely u(t) > 0 for 0<t<a .
To obtain solutions smoother than the ones usually found by his first
method, he now minimizes the expression
a ■>
\ (h'(t))^ t dt
0
c , -c
n+1 n
t ,-t

191
under the restrictions that u produces the values a. and the (physically
motivated) boundary conditions
h' @) = hA) = 0 .
The Euler-Lagrange technique leads to a nonlinear generalized eigenvalue
problem for which he gives an iterative method of computation.
9-4. A Numerical Case Study
W. Zikoll in his diploma thesis has carried out numerical case
studies using the quadratic optimization method of the preceding paragraph.
Without going into :the details of computation (let us just mention that he
used optimization algorithms of Cryer, 1971, and of Eckhardt, 1974 ),we
will display a few typical results obtained for the spectroscopic
equation (9.1.5) with a = 1, namely
(9.4.1) 2 J" u(t) tdt = f(x), 0<x<1 ,
with
(9.4.2) f(x) = jg A-x2J .
The exact solution
3/2
(9.4.3) u(t) =-1 A-t2) , 0<t<1 ,
is strictly nonotonically decreasing from 1/2 to 0 as t runs from 0 to 1 ,
concave for 0<t<1/\/2, convex for 1/\/2 < t < 1 .
As basis functions ,the hat functions (9.3.8) have been taken, the
nodes being the N equidistant points
t = n/(N-1) for n = 0,1,2,...,N-1 ,
the functionals a . are point evaluation functionals
<aj,f> = f((j-1)/J) for j = 1,2,...,J .
Actually the value J = 11 was taken.
To investigate the influence of inexact measurements of f at the
points j/J,noise was simulated by superimposing a high frequency
oscillation on f, meaning that the functionals a . have not been available on
f but rather of f + ip where
(9.4.4) cp(x) = jL sinA11-111 x ) .

192
This amounts to using (compare (9.3.2)) the values
(9.4.5) aj " IT A-x?) + <P(Xj) with x, = (j-D/J ,
j = 1,2,...,J .
We show two figures. The first one illustrates the effect of regu-
larization by dicretization, i.e. the effect of varying N, the dimension
of the space in which we are looking for an approximate solution u, the
number J - 11 of data points being kept fixed. Nonnegativity was also
used as a constraint, but the figureclearly makes visible that this
constraint alone does not have a sufficient regularizing effect.
The second figure illustrates the effect of regularization by
extra information on the shape of the solution, in one of the cases
displayed also combined with discrete Tikhonov regularization according
to (9 . 3 . 1 2) .
In both series of computations, the weight factors of (9.3.6) have
all been taken as equal to 1.

193
Fig. 9.4.1: Regularization by the number of nodes (by discretization
and nonnegativity constraint )
exact solution u
approximate solution u with perturbed data, N = 5
approximate solution u with perturbed data, N = 7
"--- "approximate" solution u with perturbed data, N = 11.
In all cases J = 11 and perturbation
1
ip(x)
10
sinA11-111 x

194
Fig. 9.4.2: Regularization by shape constraints
exact solution u
approximate solution u with perturbed data,
constraints: u(t) nonnegative and decreasing for 0<t< 1,
concave for 0<t<0.7, convex for 0.7<t<1.
In addition: Tikhonov regularization, i.e. minimization of
Qx(c) with X = 0,002 (see (9.3.12)).
approximate solution u with perturbed data. Constraint as
above.
"approximate" solution u with perturbed data, constraint:
nonnegativity.
In all cases N = J = 11 and perturbation cp(x)
1
To
sinA11-111 x

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Subject Index l
A
Abel
Abel's mechanical problem
Adams
Anderssen
a priori bounds
Arsenin
Arzela-Ascoli theorem
B
Banach
Banach space
Benndorf
Bessanova
Bessel function of first kind
Borok
Bosanquet
Branca
Brunner
C
Calligaro
Cameron
Campi
Cannon
Carleman
Cauchy principal value
Chebyshev polynomials of first kind
compact operator
compactness of Abel operator
convolution
Cormack
Cryer
crystallography
D
Dahlquist
Dirac's delta-function
Dirichlet problem
discrete probability
1
26
72
57
103
156
79
129
72
50
51
113
51
3
185
7
57
184
51
139
63
122
118
6
78
64
4
191
39
153
43
141
39
*we mention "Abel" the first time that this name appears.

211
E
Eckhardt 7
Edelstein 6
Edelstein's fixed point theorem 132
Eggermont 184
elasticity theory 2
Erdelyi 3
Erdelyi-Kober fractional operator 99
Euler beta function 11
Euler gamma function 2
evaluation of measurements 185-191
F
Fishman 51
flat earth model 51-56
Fourier transform 4
fractional derivative operator 3
fractional derivative of a composite function 98
fractional derivative of a product of functions 98
fractional integral operator 3
fractional integration by parts 98
fractional power of an operator 3
Fubini 18
G
Garmany 51
Gauss least squares fit 187
Gelfand 3
generalized Abel equations 3
generalized functions 39
geometric optics 51
Gilbert 51
Goldsmith 49
Gorenflo 188
Gram-matrix 187
Groetsh 152
Gronwall's inequality 94

212
H
Hadamard
Hairer
Han dels man
Hankel transform
Hardy
Hardy class
Herglotz
Hilbert-Riemann boundary value problem
Hilbert transform
Holder continuous spaces
Holder's inequality
Hormann
Houwen
hydrodynamics
I
ill-posed problem
integrated Lipschitz conditions
interpolating quadrature methods
inverse scattering problem for a repelling potential
inversion of seismic travel times
isochrone problem
L
Lame parameters
Laplace transform
Lark in
Lebesgue
Levinson
Levy
Linz
Liouville
Littlewood
Love
Lowengrub
Lubich
luminosity
2
152
150
4
3
105
50
63
4
69-72
67
36
185
61
6
78
184
31
50-56
12
51
63
7
17
149
188
152
3
3
3
123
149
38

213
M
Mackenzie
Mann
Marcuse
Marcinkiewicz-Zygmund interpolation theorem
McClain
McKee
mechanics
Meister
Mellin transform
metallography
Minerbo
Mittag-Leffler function
moments
Mo ran
multiplicative convolution
N
Nagumo-type condition
Natterer
Neumann initial-boundary value problem
Newton
Newtonian cooling
Newtonian heating
Newton's law of cooling
0
Olmstead
optical fibres
Orcutt
oscillating pendulum
Osier
P
Padmavally
Paley-Wiener theorem
paraxial ray equation
Parker
Parseval's relation
partial differential equations
Penzel
Peters
51
129
57
65
51
184
2
121
4
39
188
131
41
39
108
132
190
139
142
142
142
139
150
56
51
30
3
148
104
58
51
103
61
121
121

214
photoelastic effect 57
Picard 129
plasma physics 2
pressure waves 50
probability density 39
probability distribution 46
Q
quadrature methods
R
Radon transform
refractive index of optical fibres
ray parameters
regularization by discretization
Reinermann
Riele
Riemann
Riemann-Liouville fractional operator
Riesz potentials
Roberts
Runge-Kutta methods
Ryaboyi
S
Sakaljuk
Samko
scattering experiments
Schauder's fixed point theorem
Schlichte
Schwarz inequality
seismic travel times
seismic waves
seismology
shear waves
Shibata
Shilov
Shmoys
Sitnikova
Snellius refraction law
Sobolev spaces
solution formulas
spectroscopy
184-185
4
56-60
53
190
6
152
3
99
5
129
152
51
63
121
26
148
152
125
50
50
2
50
57
3
33
51
51
5
22
2

215
stability estimates 6
stability results for the general linear Abel integral equation 168
Stallbohm 6
stereology 2
stereology of spherical particles 39-49
Stieltjes integral 25
Stirling formula 167
successive approximation method 83
T
Tamark in 3
theory of elasticity 2
theory of scattering 2
Tikhonov 156
Tikhonov regularization 38
tomato salad problem 48
tomography of the earth 51
Tonelli 3
trapezoidal product integration method 184
travel time curve 56
Tricomi's equation 62
V
van der Houwen 7
Volterra 2
Volterra integral equations 2
W
Walton 173
weak singularity 2
well-posed 49
Weyl 3
Weyl fractional operator 99
Wiechert 50
Wiener-Hopf equation 4
Wolf 139
Y
Young 3
Young's inequality 65
Z
Zikoll
7

Lecture Notes in Mathematics
Edited by A. Dold, B. Eckmann and F. Takens
Editorial Policy
for the publication of monographs
In what follows references to monographs, are applicable also to multiauthorship
volumes such as seminar notes.
§1. Lecture Notes aim to report new developments - quickly, informally, and at a
high level. Monograph manuscripts should be reasonably self-contained and
rounded off. Thus they may, and often will, present not only results of the author
but also related work by other people. Furthermore, the manuscripts should
provide sufficient motivation, examples and applications. This clearly distinguishes
Lecture Notes manuscripts from journal articles which normally are very
concise. Articles intended for a journal but too long to be accepted by most journals,
usually do not have this "lecture notes" character. For similar reasons it is unusual
for Ph.D. theses to be accepted for the Lecture Notes series.
Experience has shown that English language manuscripts achieve a much wider
distribution.
§2. Manuscripts or plans for Lecture Notes volumes should be submitted (preferably
in duplicate) either to one of the series editors or to Springer- Verlag, Heidelberg.
These proposals are then refereed. A final decision concerning publication can
only be made on the basis of the complete manuscript, but a preliminary decision
can often be based on partial information: a fairly detailed outline describing the
planned contents of each chapter, and an indication of the estimated length, a
bibliography, and one or two sample chapters - or a first draft of the manuscript.
The editors will try to make the preliminary decision as definite as they can on the
basis of the available information.
§3. Final manuscripts should contain at least 100 pages of mathematical text and
should include
- a table of contents;
- an informative introduction, perhaps with some historical remarks: it should
be accessible to a reader not particularly familiar; with the topic treated;
- a subject index: as a rule this is genuinely helpful for the reader.

General Remarks
Lecture Notes are printed by photo-offset from the master-copy delivered in
camera-ready form by the authors of monographs, resp. editors of proceedings
volumes. For this purpose Springer-Verlag provides technical instructions for the
preparation of manuscripts. Volume editors are requested to distribute these to all
contributing authors of proceedings volumes. Some homogeneity in the presentation of the
contributions in a multi-author volume is desirable.
Careful preparation of manuscripts will help keep production time short and ensure a
satisfactory appearance of the finished book. The actual production of a Lecture Notes
volume normally takes approximately 8 weeks.
For monograph manuscripts typed or typeset according to our instructions,
Springer-Verlag can, if necessary, contribute towards the preparation costs at a fixed
rate.
Authors of monographs receive 50 free copies of their book. Editors of proceedings
volumes similarly receive 50 copies of the book and are responsible for redistributing
these to authors etc. at their discretion. No reprints of individual contributions can be
supplied. No royalty is paid on Lecture Notes volumes.
Volume authors and editors are entitled to purchase further copies of their book for their
personal use at a discount of 33.3 %, other Springer mathematics books at a discount of
20 % directly from Springer-Verlag. Authors contributing to proceedings volumes may
purchase the volume in which their article appears at a discount of 20 %.
Commitment to publish is made by letter of intent rather than by signing a formal
contract. Springer-Verlag secures the copyright for each volume.
Addresses:
Professor A. Dold, Mathematisches Institut, Universitat Heidelberg,
Im Neuenheimer Feld 288, 6900 Heidelberg, Federal Republic of Germany
Professor B. Eckmann, Mathematik, ETH-Zentrum
8092 Zurich, Switzerland
Professor F. Takens, Mathematisch Instituut, Rijksuniversiteit Groningen,
Postbus 800, 9700 AV Groningen, The Netherlands
Springer-Verlag, Mathematics Editorial, Tiergartenstr. 17,
6900 Heidelberg, Federal Republic of Germany, Tel.: *49 F221) 487-410
Springer-Verlag, Mathematics Editorial, 175 Fifth Avenue,
New York, New York 10010, USA, Tel.: *1 B12) 460-1596