Ebook: Fractional calculus and integral transforms of generalised functions

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.