The purpose of this book is to give a self-contained exposition of the geometric theory of Bochner-Riesz means. The subject deals with the most basic topic in Fourier analysis, the question of when a Fourier series converges to its original function. Substantial progress was made in the mid 1970's, but the techniques are still avaliable only in the technical literature. Our intent is to present an account accessible to graduate students. We have slighted certain important topics in order to maintain a consistent presentation. We have assumed that the reader is familiar with real analysis at a graduate level, and with basic facts about distributions and the Fourier transform. A basic reference is the text by Stein and Weiss, Introduction to Fourier Analysis on Euclidean Spaces [50], and the texts of Rudin, [43] and [43].

In writing this book, we benefitted with extensive conversations over many years with our colleagues. We wish to thank Professors E. Fabes, R. Fefferman, and E. M. Stein for their help with the material in Chapters 1, 2 and 3. The contents of Chapters 4 and 5 were influenced by conversations with Professor A. W. Knapp. For the general philosophy of Chapters 7 and 8 we are indebted to Professors A. Cordoba and C. Fefferman. The first draft of this book was written while the first author was supported by NSF grants MCS 8202165 and 8001799.