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First published in 1968, An Introduction to Harmonic Analysis has firmly established itself as a classic text and a favorite for students and experts alike. Professor Katznelson starts the book with an exposition of classical Fourier series. The aim is to demonstrate the central ideas of harmonic analysis in a concrete setting, and to provide a stock of examples to foster a clear understanding of the theory. Once these ideas are established, the author goes on to show that the scope of harmonic analysis extends far beyond the setting of the circle group, and he opens the door to other contexts by considering Fourier transforms on the real line as well as a brief look at Fourier analysis on locally compact abelian groups. This new edition has been revised by the author, to include several new sections and a new appendix.
Harmonic analysis is the study of objects (functions, measures, etc.), defined on topological groups. The group structure enters into the study by allowing the consideration of the translates of the object under study, that is, by placing the object in a translation-invariant space. The study consists of two steps. First: finding the “elementary components” of the object, that is, objects of the same or similar class, which exhibit the simplest behavior under translation and which “belong” to the object under study (harmonic or spectral analysis); and second: finding a way in which the object can be construed as a combination of its elementary components (harmonic or spectral synthesis).
The vagueness of this description is due not only to the limitation of the author but also to the vastness of its scope. In trying to make it clearer, one can proceed in various ways; we have chosen here to sacrifice generality for the sake of concreteness. We start with the circle group T and deal with classical Fourier series in the first five chapters, turning then to the real line in Chapter VI and coming to locally compact abelian groups, only for a brief sketch, in Chapter VII. The philosophy behind the choice of this approach is that it makes it easier for students to grasp the main ideas and gives them a large class of concrete examples which are essential for the proper understanding of the theory in the general context of topological groups.