Fourier transforms. Principles and applications (Q2925811)

Of course, my goal is to write a more or less standard review, however, very often my personal feelings and impressions replace formal description. It is a very special thing to read a book on the subject the reader works on all his life and to feel as if being on a masquerade. Familiar things and familiar titles appear then in an unusual and unexpected form. Basics of the author's approach are described immediately on the second page of the preface: ``The four types of Fourier transform on discrete and continuous domains -- discrete Fourier transform (DFT), Fourier series, discrete-time Fourier transform (DTFT), and Fourier transform -- are developed as orthogonal expansions within a vector space framework.'' I never felt any need in four types of Fourier transforms and continue to feel that. Therefore I prefer to briefly describe each of the ten chapters.NEWLINENEWLINEChapter 1. Review of prerequisite mathematics. In this chapter very rudimental facts are given, as if recalling well forgotten basics. Chapter 2. Vector spaces. This chapter is somewhat similar to the previous one but on a much more serious level of functional analysis. Chapter 3. The discrete Fourier transform. This chapter is a core, in a sense, since the author builds all the other transforms on the basis of the discrete Fourier transform. Chapter 4. The Fourier series. I was pleased to read in this ``applications'' book Kolmogorov's result on the existence of the Fourier series of an integrable function divergent everywhere, even without mentioning Kolmogorov. However, the latter is pity, since, in my opinion, every possibility to incorporate history should be used in textbooks. What is strange to me is how the author avoided the notion of bounded variation in a more than 750 pages book. The other similar issue is summability. Of course,``the DFT may be used to compute approximations to the Fourier series and the DTFT'', but most of good approximations are built on the basis of summability. Chapter 5. The Fourier transform. Similar words can be said on this chapter. What is of special interest is that summability is widely used here, but only by means of a special, Gaussian method. Chapter 6. Generalized functions. Introducing such a chapter in a book for applications is always disputable. I prefer it to be here, no matter how much justified. Chapter 7. Complex function theory and Chapter 8. Complex integration. What can I say about these chapters and their place? Only that the author is sure that this is a proper moment to introduce these important notions. Chapter 9. Laplace, Z, and Hilbert transforms. Of course, these transforms are important for applications. I was especially interested in reading about the Hilbert transform. I dare suppose that the problem of how to introduce a conjugate function for a function just bounded should appear in that context. Chapter 10. Fourier transforms in two and three dimensions. In fact, most of the formulas are given in a general form, while examples and applications are in dimensions two and three. Such an approach definitely seems reasonable.NEWLINENEWLINEIt is convenient that every chapter ends up with a summary of the results considered and a bunch of exercises. I hope the author's experience and expertise are what had inspired him to write this book of the present form, size and choice of matter. I also hope that it will find additional readers beyond the author's students.