zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 ' , 𝒮 ' and 𝒟 ' . 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:
46FxxDistributions, generalized functions, distribution spaces
46-01Textbooks (functional analysis)
47G30Pseudodifferential operators
42C15General harmonic expansions, frames
42A38Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type