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)
Fourier analysis. An Introduction. (English) Zbl 1026.42001
Princeton Lectures in Analysis. 1. Princeton, NJ: Princeton University Press. xvi, 311 p. $49.95; £ 35.00 (2003). This book is an excellent introduction to the Fourier analysis. It is the first volume of the four planned volumes based on a series of four one-semester courses taught at Princeton University whose purpose was to present, in an integrated manner, the core areas of analysis. As the authors write: “The objective was to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science.” The reviewer is in favour of the concept of the authors given in the Preface to Volume I: “Not to overburden the beginning student with some of the difficulties that are inherent in the subject: a proper appreciation of the subtleties and technical complications that arise can come only after one has mastered some of the initial ideas involved. This point of view has led us to the following choice of material in the present volume: {\it Fourier series}. At this early stage it is not appropriate to introduce measure theory and Lebesgue integration. For this reason our treatment of Fourier series in the first four chapters is carried out in the context of Riemann integrable functions.… {\it Fourier transform}. For the same reasons, instead of undertaking the theory in a general setting, we confine ourselves in Chapter 5 and 6 largely to the framework of test functions.… {\it Finite Fourier analysis}. This is an introductory subject {\it par excellence}, because limits and integrals are not explicitly present. Nevertheless, the subject has several striking applications, including the proof of the infinitude of primes in an arithmetic progression.” The prerequisites are kept to a minimum, only some acquaintance with the notion of the Riemann integral is supposed, but an appendix contains most of the results needed in the text. For further orientation here is the list of chapter headings: The Genesis of Fourier Analysis, Basic Properties of Fourier Series, Convergence of Fourier Series, Some Applications of Fourier Series, The Fourier Transform on$\bbfR$, The Fourier Transform on$\bbfR^d\$, Finite Fourier Analysis, Dirichlet’s Theorem. Furthermore there are Appendix, Notes and References, Bibliography, Symbol Glossary. Each chapter ends with Exercises and Problems. If the reader solves these problems, some of them are very hard in spite of the useful hints, he gets a very good practice in analysis. The reviewer knows, what the authors hope, that their approach will facilitate their goal: “To inspire the interested reader to learn more about this fascinating subject, and to discover how Fourier analysis affects decisively other parts of mathematics and science.” We warmly recommend this valuable work to everybody, from students of mathematics, physics, engineering and other sciences, to teachers and mathematicians, who is interested in Fourier series.

MSC:
 42-01 Textbooks (Fourier analysis) 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42B10 Fourier type transforms, several variables
Keywords:
textbook; Fourier analysis