Fractional calculus and integral transforms of generalised functions

✍ Scribed by McBride A. C.

• Publisher: Pitman
• Year: 1979
• Tongue: English
• Leaves: 190
• Series: Research notes in mathematics 31
• Category: Library

✦ Synopsis

This book is concerned with the study of certain spaces of generalized functions and their
application to the theory of integral transforms defined on the positive real axis. Dr. McBride has
purposely chosen to study only a few operators in considerable detail rather than hurriedly
rushing over a larger number of transforms for which his results are applicable, and has used as a
unifying theme the operators of fractional integration. A major contribution of the author is to
construct spaces of generalized functions which are applicable to several different operators at
the same time instead of having to change spaces each time the operator is changed. This is of
crucial importance in almost all cases of practical importance since in such cases it is usually
necessary to apply a succession of operators in order to arrive at a solution. The plan of the book
is as follows. In Chapter Two the basic spaces of testing functions and generalized functions are
introduced and their algebraic and topological properties studied. Chapter Three is devoted to the
development of the operators of fractional integration defined on the previously studied space of
generalized functions and in Chapter Four these results are applied to certain integral equations
having a hypergeometric function as the kernel. Chapters Five and Six are concerned with the
Hankel transform defined on spaces of generalized functions and the close connections existing
between this transform and fractional calculus. Chapter Seven is in a sense the highlight of the
book where the material in Chapters Three, Five and Six is applied to the study of dual integral
equations of Titchmarsh type. In particular, the author is able to establish the existence and
uniqueness of classical solutions to such a system. The uniqueness part of the argument is
particularly elegant and employs the full power of the previously developed theory. Finally, in
Chapter Seven, the author briefly indicates how his methods can be used to study other classes of
integral operators defined on (0, <»).
Dr. McBride has made strenuous efforts to develop his theory as concisely as possible and not
to wander off in tangential directions. His aim of showing "how the general theory incorporates
the classical theory and, at the same time, provides a framework wherein the formal analysis
found in many books and papers can be justified rigorously" has been admirably fulfilled in a
clear and lively style. The author has stated in his preface his hope that this book might serve as a
modest tribute to his thesis advisor, the late Professor Arthor Erdelyi. In terms of subject matter,
significance of results, and excellent use of the English language, the book of Dr. McBride fulfills
the highest standards set by his mentor and his "modest tribute" is in fact a major contribution to
the area of mathematics in which Professor Erdelyi devoted much of his mathematical life. It
should occupy a prominent place on the bookshelf of every mathematician interested in classical
analysis and its applications.

✦ Table of Contents

McBride A.C.Fractional calculus and integral transforms of generalised functions (Research notes in mathematics,31)( Pitman,1979)(ISBN 0273084151)(600dpi)(190p) ......Page 4
Copyright ......Page 5
Contents ......Page 8
Preface ......Page 6
0 Notation 1 ......Page 10
1.1 Motivation and Background 4 ......Page 13
1.2 Plan of Campaign 12 ......Page 21
2.1 Definition and Simple Properties of the Spaces F(p,m) 14 ......Page 23
2.2 Simple Operators on the F(p,m) Spaces 21 ......Page 30
2.3 The Spaces F1 of Generalised Functions 28 ......Page 37
2.4 Simple Operators in F1 29 ......Page 38
3.1 Introduction 36 ......Page 45
3.2 Fractional Calculus in F(p,m) : Stage I 40 ......Page 49
3.3 Fractional Calculus in F(p,m) : Stage II 57 ......Page 66
3.4 Some Consequences 68 ......Page 77
3.5 Definition of the Operators in F1 73 ......Page 82
3.6 A Simple Application 80 ......Page 89
4.1 Introduction 87 ......Page 96
4.2 The Operators H.(a,b;c;m) on F(p,m) 89 ......Page 98
4.3 The Operators H.(a,b;c;m) on F1 96 ......Page 105
4.4 The Classical Case 98 ......Page 107
5.1 Introduction 103 ......Page 112
5.2 H on F(p,m) : The Simplest Case 105 ......Page 114
5.3 H on F(p,m) : The Extended Operator 110 ......Page 119
5.4 H on F1 118 ......Page 127
5.5 The Modified Hankel Transform S(n,a) 120 ......Page 129
6.2 Erdelyi-Kober Operators and H. 123 ......Page 132
6.3 Erdelyi-Kober Operators and S(n,a) 129 ......Page 138
7.1 Introduction 133 ......Page 142
7.2 A Technicality 135 ......Page 144
7.3 Existence of Solutions of Problem 7.3 137 ......Page 146
7.4 Uniqueness for Problem 7.3 140 ......Page 149
7.5 Uniqueness for Problem 7.1: The Simplest Case 146 ......Page 155
7.6 Uniqueness for Problem 7.1: Other Cases 155 ......Page 164
8.2 Other Transforms 158 ......Page 167
8.3 Modifications of the Spaces 164 ......Page 173
8.4 Subspaces of F(p,m) 168 ......Page 177
8.5 Concluding Remarks 171 ......Page 180
References 173 ......Page 182
cover ......Page 1
back cover ......Page 190