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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.

42-01Textbooks (Fourier analysis)
42A38Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B10Fourier type transforms, several variables