This book provides a self-contained treatment of classical Fourier analysis and a concrete introduction to a number of topics in harmonic analysis at the upper undergraduate or beginning graduate level. Necessary prerequisites to using the text are rudiments of the Lebesgue measure and integration on the real line. Frequently, more than one proof is offered for a given theorem to illustrate the multiplicity of approaches. The text contains numerous examples and more than 175 exercises that form an integral part of the text. It can be expected that a careful reader will be able to complete all these exercises. The book contains six chapters. The first chapter provides a reasonably complete introduction to Fourier analysis on the circle and its applications to approximation theory, probability and plane geometry (the isoperimetric theorem). Readers with some sophistication but little previous knowledge of Fourier series can begin with chapter 2 and anticipate a self-contained treatment of the n-dimensional Fourier transform and many of its applications.
Much of modern harmonic analysis is carried out in the Lp spaces for p ≥ 2, which is the subject of chapter 3. The Poisson summation formula treated in this chapter provides an elegant connection between Fourier series on the circle and Fourier transforms on the real line, culminating in the Landau asymptotic formulas for lattice points on a large sphere. This chapter also contains the interpolations theorems of Riesz-Thorin and Marcinkiewicz, which are applied to discuss the boundedness of the Hilbert transform and its application to the Lp convergence of Fourier series and integrals.
In chapter 4 the author merges the subjects of Fourier series and Fourier transforms by means of the Poisson summation formula in one and several dimensions. Chapter 5 explores an application of Fourier methods to probability theory. The central limit theorem, the iterated log theorem and Berry-Esseen theorems are developed using the suitable Fourier-analytic tools. The final chapter furnishes a gentle introduction to wavelet theory, depending only on the L2 theory of the Fourier transform (the Plancherel theory). The basic notions of scale and location parameters demonstrate the flexibility of the wavelets approach to harmonic analysis. The book can be used as a text for courses, seminars or even solo study and is designed for graduate students in mathematics, physics, engineering and computer science.