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Introduction to Fourier series. (English) Zbl 0848.42001

Pure and Applied Mathematics, Marcel Dekker. 199. New York, NY: Marcel Dekker. vii, 285 p. (1996).
The concept of Fourier series is a fundamental tool in Analysis. Although there exist at present more than a dozen books, mainly monographs, on the theory of Fourier series, there has to date been a certain gap in the textbook literature.
The main aim of this book is to provide an introduction to some aspects of Fourier series and related topics, in which a liberal use is made of modern techniques. This emphasis affects not only the type of arguments, but also to certain extent the choice of material. Among others, the idea of homogeneous Banach spaces is stressed throughout the book, the concept of approximate identities plays a basic role in the treatment, and a substantial portion of Approximation Theory is included, where the partial sums or various means of Fourier series are used to approximate a function. Interesting applications are also made, for example, for weakly stationary stochastic processes in prediction theory, or the Whittaker-Shannon-Kotel’nikov sampling theorem is derived on \(\mathbb{R}\), and more.
The table of contents is the following: 1. Fourier coefficients, 2. Approximate identities, 3. Approximate identities and pointwise convergence, 4. Square integrable functions, 5. Convergence of Fourier series in norm, 6. Local convergence, 7. Characterization of Fourier coefficients, 8. Hilbert transform, 9. Characterization of approximate identities, 10. Triangular schemes, 11. Elements of best approximation, 12. Poisson integrals and Hardy spaces, 13. Conjugation of approximate identities, 14. Szegö-Kolmogorov theorem, 15. Absolute convergence of Fourier series, 16. Fourier transform on \(\mathbb{R}\), 17. Plancherel transform on \(\mathbb{R}\), 18. Poisson summation formula.
Each chapter ends with Exercises, the more difficult ones being provided with hints to their solutions. The book begins with Preface, and ends with Appendices, References, and Index.
The reader is supposed only to be familiar with Lebesgue integration and some elements of Functional Analysis. For convenience, the basic notions and theorems are collected in the Appendices.
The author’s guiding principle was to provide an introductory, self-contained, concise treatise on Fourier series, while making use of some techniques of Functional Analysis and Measure Theory. And indeed, this book is an excellent introduction. It is addressed to undergraduate students and warmly recommended to everyone who wants to make a quick acquaintance with Fourier series.
Reviewer: F.Móricz (Szeged)

MSC:

42-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
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