Krantz S. A Panorama of Harmonic Analysis

Washington: The Mathematical Association of America, 1999. — 375 p.The purpose of the present book is to give the uninitiated reader an historical overview of the subject of Fourier analysis as we have just described it. While this book is considerably more polished than merely a set of lectures, it will use several devices of the lecture: to prove a theorem by considering just an example; to explain an idea by considering only a special case; to strive for clarity by not stating the optimal form of a theorem.
We shall not attempt to explore the more modern theory of Fourier analysis of locally compact abelian groups (Le., the theory of group
characters), nor shall we consider the Fourier analysis of non-abelian groups (i.e., the theory of group representations). Instead, we shall restrict attention to the classical Fourier analysis of Euclidean space.
Prerequisites are few. We begin with a quick and dirty treatment of the needed measure theory and functional analysis. The rest of the book uses only elementary ideas from undergraduate real analysis. When a sophisticated idea is needed, it is quickly introduced in context.
It is hoped that the reader of this book will be imbued with a sense of how the subject of Fourier analysis has developed and where it is heading. He will gain a feeling for the techniques that are involved and the applications of the ideas. Even those with primary interests in other parts of mathematics should come away with a knowledge of which parts of Fourier analysis may be useful in their discipline, and also where to tum for future reading.
In this book we indulge in the custom, now quite common in harmonic analysis and partial differential equations, of using the same letters (often C or C' or K) to denote different constants-even from line to line in the same proof. The reader unfamiliar with this custom may experience momentary discomfort, but will soon realize that the practice streamlines proofs and increases understanding.