Principles of Fourier analysis (Q2832099)

The book under review is the second edition of the book from 2001 that is reviewed in [Zbl 1022.42001]. As it is noted in the preface, it is a book on the mathematics of the Fourier analysis, and not a book on its applications. We recall from [loc. cit.] that the author introduces the approach based on the Gaussian test functions (Chapter IV). This provides a generalized theory whose special cases are the classical (Chapter III) and discrete (Chapter V) theories.NEWLINENEWLINEThe book is organized in five parts, as its first edition, with the following additional material. Exercises 11.11 and 11.12 related to the Haar wavelet, Section 14.4 on divergent Fourier series, a brief discussion of the classical sampling theorem for band-limited functions in Chapter 20, Chapter 28 on multi-dimensional Fourier transform and Section 29.6 relating the Fourier and Laplace transforms.NEWLINENEWLINEThe enumeration of the chapters is the same as in the previous edition with the following changes. Chapters 26, 27, 28 and 29 of the first edition are now Chapters 29, 26, 27 and 30, respectively. Chapters 30--39 of the first edition are now Chapters 31--40, respectively.NEWLINENEWLINEThis voluminous book may well serve as the main text for a sequence of one-term courses on Fourier analysis. The author offers a proposal of two possible single-semester courses of the introductory and intermediate level.NEWLINENEWLINEThe exposition is narrative and contains numerous well distributed explanations, comments and remarks. This is in accordance to one of the author's overriding goals stated in the preface: ``intelligent employment'' of Fourier analysis in the sense of ``understanding what formulas are really saying, when they can be used, how they can be used, and how they should not be used.''