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A guide to distribution theory and Fourier transforms. (English) Zbl 0854.46035
Studies in Advanced Mathematics. Boca Raton, FL: CRC Press. x, 213 p. \$ 49.95 (1994).

This book presents in a concise and polished form the basic properties of the subject. The use of topological vector spaces and measure theory is avoided. A treatment of microlocal analysis in texts of this kind is new.

The book is largely self-contained. For reading it only the knowledge of the multi-dimensional calculus and some complex analysis is assumed. It is written carefully in a relaxed, friendly and teaching style (at some time figurative). Its forte feature is an illuminating and analytic explanation of numerous points of the text.

The first two chapters (“What are distributions?” and “Fourier transforms”) motivate and introduce the basic concepts and computational techniques. Chapters three and four do the same for Fourier transforms. Chapter five solves a few classical equations that arise in mathematical physics by taking Fourier transforms. Chapter six (“The structure of distributions”) explains the notion of continuity and states structure theorems for the spaces ${ℰ}^{\text{'}}$, ${𝒮}^{\text{'}}$ and ${𝒟}^{\text{'}}$. Chapter seven (“Fourier analysis”) provides the Paley-Wiener theorems, Poisson summation formula, proof of the Heisenberg uncertainty principle, Haar functions and wavelets. The last chapter is devoted to Sobolev embedding theorems, Sobolev spaces, equations of elliptic and hyperbolic types, pseudodifferential operators, the wave front set and microlocal analysis.

Each chapter is furnished (at the end) with a number of selected problems of varying levels of difficulty. Some of them extend the present material. In all there are 211 problems. Some examples and exercises are distributed into the chapters. The book closes with an index.

We appreciate the author’s “Suggestions for further reading”, but a bibliography on distribution theory and the line microlocal analysis – pseudodifferential operators, and wavelets, would be more complete.

The reviewer found the book enjoyable to read and appropriate as a text for an introductory advanced undergraduate or graduate course (in applied mathematics).

##### MSC:
 46Fxx Distributions, generalized functions, distribution spaces 46-01 Textbooks (functional analysis) 47G30 Pseudodifferential operators 42C15 General harmonic expansions, frames 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type