Fourier analysis. Part I: Theory (Q5890781)

Many books on the basics of Fourier Analysis appeared during last years. The author does not refer to most of them and naturally does not compare the work under review with them. However a natural question appears what differs this book from its predecessors. Roughly, what can be said is that the author tries to tell the reader commonly known things in his own words. In the preface, it is said that ``In writing this textbook the author has acted primarily as a reporter, not a researcher''. Well, it can be said that maybe, for all that, also as a researcher of how to report the matter. It is somewhat early and maybe not completely fair to judge whether the goal is achieved, since this is the first volume from the two volumes textbook. The second volume where the applications are given will probably show how good the choice of theoretical matter is and how the manner this matter is given helps to deal with applications. However, what can already be mentioned is that numerous exercises given along with hints and solutions is a positive feature of the book. Though only the basics of the theory are given, the author is not going to ``spare'' his readers by avoiding certain aspects of analysis, like the Lebesgue measure and integration, certain facts of functional analysis, and some other advanced and rather modern issues. To see these in detail, let us go along the contents, chapter by chapter.NEWLINENEWLINEThe first one, introductive, goes back to the origins of the theory. We mention immediately that at the end of the book, in Appendix, a list of those who contributed to it during two centuries is given, with dates of life and brief description of their activities. Also, all the chapters (but the last one, which is just a ``glance'') are accompanied by historical and bibliographical notes. The second chapter presents essentials of the Lebesgue measure and integration. In the next chapter, the basics of functional analysis are given in the same brief manner, however touching the main aspects of it. Only Chapter 4 starts with ``genuine'' Fourier Analysis by dealing with the problems of convergence of Fourier series. Though the word ``summability'' cannot be found in the index, certain aspects of it are considered. Chapter 5 concerns the Fourier transforms, in dimension one, as in the previous chapter, while Chapter 6 presents more or less the same issues as in the two preceding chapters, but in the multivariate setting. It also touches such topics as distributions, positive definiteness, interpolation, Sobolev spaces, etc. The last (in this volume) Chapter 7 entitled ``A glance at some advanced topics'' presents, in fact, in a brief but very dense manner all the basics of complex analysis. Through the Paley-Wiener theorems (in one and several dimensions) and Hardy spaces it is related to Fourier Analysis.NEWLINENEWLINEOf course, there will be some groups of students to whom this book will be too difficult or too easy, but it is even more definite that there are many groups such that this book will fit both the students and their instructors. The amount of such courses as a sign of the success of the book under review will apparently be revealed after the second volume appears and is used. The author himself wishes he had the material of this book before beginning his own graduate courses.