Commutative Harmonic Analysis I

This is the first volume of four already scheduled in the subseries Commutative Harmonic Analysis of the EMS. Starting from the level of the average graduate student, it ends with some topics in modern Fourier analysis. Besides its primary audience, the book should be useful to researchers in neighbouring fields, e.g. ODE, PDE, mathematical physics.

Klappentext

This is the first volume in the subseries Commutative Harmonic Analysis of the EMS. It is intended for anyone who wants to get acquainted with the discipline. The first article is a large introduction, also serving as a guide to the rest of the volume. Starting from Fourier analysis of periodic function, then going through the Fourier transform and distributions, the exposition leads the reader to the group theoretic point of view. Numer- rous examples illustrate the connections to differential and integral equations, approximation theory, number theory, probability theory and physics. The article also contains a brief historical essay on the development of Fourier analysis. The second article focuses on some of the classical problems of Fourier series; it's a "mini-Zygmund" for the beginner. In particular, the convergence and summability of Fourier series, translation invariant operators and theorems on Fourier coefficients are given special attention. The third article is the most modern of the three, concentrating on the theory of singular integral operators. The simplest such operator, the Hilbert transform, is covered in detail. There is also a thorough introduction to Calderon-Zygmund theory.

Inhalt
I. Methods and Structure of Commutative Harmonic Analysis.- II. Classical Themes of Fourier Analysis.- III. Methods of the Theory of Singular Integrals: Hilbert Transform and Calderón-Zygmund Theory.- Author Index.