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Introduction to Transform Theory

David V. Widder
Academic Press
Publication Date: 
Number of Pages: 
Electronic Book
Pure and Applied Mathematics
[Reviewed by
Allen Stenger
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This book gives a concise introduction to the most important facts about integral transforms, aimed at senior undergraduate or first-year graduate students. Prerequisites for the book are a good course in advanced calculus; Widder references his own Advanced Calculus for most of the facts used here. The book includes exercises at the end of each chapter, but they are not very challenging; most are to work out specific examples or to tie up loose ends from the chapter.

The book is not a survey of integral transforms: it deals primarily in the Laplace transform, but also deals with several analogous transforms and shows how the ideas and properties of Laplace transforms can be applied more widely. It also covers the general convolution transform, which is simply the convolving of a given kernel with the desired function, and shows how much can be proved about these transforms with minimal knowledge of the kernel. It does not deal with Fourier transforms except to develop a little bit of Fourier theory in support of some of the topics. This book is just an introduction, and Widder also wrote a number of monographs that cover these transform topics in much more detail.

An unusual feature of the book, that works very well, is that the exposition starts out with general Dirichlet series, which are presented as the discrete form of the Laplace transform. This allows the main theorems to be developed in a simpler environment, and then we can guess the forms of the analogous theorems for the integral transforms. The Dirichlet series material is fairly in-depth and takes up about a third of the book. It includes a self-contained proof of the Prime Number Theorem that foreshadows some of the analyticity and growth theorems and Tauberian theory.

Another feature I especially appreciate is that, even though it is a pure-math book, the first chapter develops quickly, in a cookbook fashion, enough Laplace transform material to show how they can be used to solve ordinary differential and integral equations. As the author says (p. ix), “any student of transform theory will wish to be cognizant of this most important application.”

The book also has the beginnings of the moment problem, quite a bit on inverting transforms (both through complex variable and through real variable methods), the representation problem (which properties guarantee that a function can be represented as a particular transform), and Tauberian theorems.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

Dirichlet Series
The Zeta Function
The Prime Number Theorem
The Laplace Transform
Real Inversion Theory
The Convolution Transform
Tauberian Theorems
Inversion by Series