Absolutely convergent Fourier series, classical function classes and Paley’s theorem

This is a survey paper on the recent progress in the study of the continuity and smoothness properties of a function f with absolutely convergent Fourier series. We give best possible sufficient conditions in terms of the Fourier coefficients of f which ensure the belonging of f either to one of the Lipschitz classes Lip(α) and lip(α) for some 0 < α ≤ 1, or to one of the Zygmund classes Zyg(α) and zyg(α) for some 0 < α ≤ 2. We also discuss the termwise differentiation of Fourier series. Our theorems generalize those by R. P. Boas Jr., J. Németh and R. E. A. C. Paley, and a number of them are first published in this paper or proved in a simpler way.