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Ferenc Móricz

University of Szeged (Hungary)

Absolutely convergent Fourier series and classical function classes

This is a survey talk 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 in order that f belong 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. The termwise differentiation of Fourier series is also discussed. Our theorems generalize some of those proved by R.P. Boas Jr., J. Németh and R.E.A.C. Paley.