Trigonometrical series. (English) Zbl 0011.01703

Zygmund, Antoni

Monografie Matematyczne 5. Warszawa: Seminarium Matematyczne Uniwersytetu Warszawskiego; Warszawa: Instytut Matematyczny PAN; Warszawa-Lwow: Subwencji Funduszu Kultury Narodowej. 331 S. (1935).

A thoroughly modern and remarkably complete account of the theory of trigonometric series. The chapter headings follow with running comments.
I. Trigonometrical series and Fourier series. II. Fourier coefficients. Tests for the convergence of Fourier series. III. Summability of Fourier series (chiefly Abel and Cesàro). IV. Classes of functions and Fourier series. (Conditions that a Fourier series belongs to a function of given class, factor sequences, Riesz-Fischer theorem, Parseval’s relations.) V. Properties of some special series. (Monotone coefficients, dispersion power series, lacunary series, series in Rademacher’s functions.) VI. The absolute convergence of trigonometrical series (structure of convergence set, sufficient conditions, Wiener’s theorem). VII. Conjugate series and complex methods in the theory of Fourier series (chiefly theorems of M. Riesz and related questions). VIII. Divergence of Fourier series (including Kolmogoroff’s example). Gibbs’s phenomenon. IX. Further theorems on Fourier coefficients. Integration of fractional order. (Young-Hausdorff theorem, M. Riesz’ convexity theorems, rearrangement theorems of Hardy-Littlewood-Paley, lacunary series.) X. Further theorems on the summability and convergence of Fourier series. (Strong summability, the Kolmogoroff-Seliverstoff theorem, conditions for C-summability.) XI. Riemann’s theory of trigonometrical series (principle of localization, sets of uniqueness, results of Rajchmann.) XII. Fourier integrals \((1\le p\le 2)\).
Every chapter is completed by an interesting collection of miscellaneous theorems and examples. Large bibliography.
The excellent treatise contains a number of important original contributions by the author.

Reviewer: Einar Hille (New Haven, Conn.)

MSC:

42-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
42Axx	Harmonic analysis in one variable

Keywords:

trigonometric series