Harmonic analysis. From Fourier to wavelets (Q2881013)

This book is mainly addressed to advanced undergraduate and beginning graduate students. It consists of twelve chapters preceded by a preface and ended by an appendix. The first five chapters are devoted to Fourier series. Starting from basic notions and physical motivations pointwise and mean convergence of Fourier series and summability methods are discussed. In Chapter 6 finite Fourier analysis is examined including the FFT algorithm. In Chapter 7 and 8 the Fourier transform on the line is discussed, the Schwartz class is introduced and the theory of tempered distribution is presented. Also a few canonical applications of the Fourier transform, including the Shannon sampling theorem, are surveyed. In Chapter 9, 10 and 11 wavelet bases are discussed, with emphasis on the Haar basis. Discussion starts with windowed Fourier transform, the Gabor transform and the wavelet transform, continues through the general framework of multiresolution analysis and description of some typical applications to image processing, and ends with description of properties and design features of known wavelets. Finally, in Chapter 12 the theory of the Hilbert transform is presented and connections of the Hilbert transform with complex analysis are exhibited.NEWLINENEWLINEThere are numerous exercises dispersed throughout the book, some of them invite a potential reader to prove stated theorems or their analogues. Each chapter ends with the so-called `projects' (altogether there are 24 of them and they vary in difficulty and sophistication) aimed at individual students or teams of students using the book. The idea of projects looks really interesting in the age of Wikipedia and search engines.NEWLINENEWLINEA number of theorems, especially in the introductory Chapters 2--5, is left without proofs; then the authors provide clear pointers to where the proofs may be found (included are classical sources like \textit{R. S. Strichartz} [The way of analysis. Boston, MA: Jones and Bartlett Publishers (1995; Zbl 0878.26001)] and \textit{T. W. Körner} [Fourier analysis. Cambridge etc.: Cambridge University Press (1988; Zbl 0649.42001)]). Some parts of Chapters 10 and 11 closely follow material from the first author's book \textit{M. J. Mohlenkamp} and \textit{M. C. Pereyra} [Wavelets, their friends, and what they can do for you. EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (2008; Zbl 1153.42016)]. Also in Chapter 10 frequent references to the monograph \textit{P. Wojtaszczyk} [A mathematical introduction to wavelets. London Mathematical Society Student Texts. 37. Cambridge: Cambridge University Press (1997; Zbl 0865.42026)] are given.NEWLINENEWLINEThe text is accompanied by a number of illuminating figures. It is also worth emphasizing that the panorama of harmonic analysis presented in the book includes very recent achievements like the connection of the dyadic shift operator with the Hilbert transform. This gives to an interested reader a good chance to see concrete examples of contemporary research problems in harmonic analysis.NEWLINENEWLINEI highly recommend this book as a good source for undergraduate and graduate courses as well as for individual studies.