On certain sums of Fourier-Stieltjes coefficients

Abstract

We obtain estimates for certain sums of Fourier-Stieltjes (and hence also Fourier) coefficients of continuous functions f f of bounded variation in terms of the modulus of continuity of f f . As a consequence of one of our results we obtain an improvement on a theorem of Zygmund on the absolute convergence of Fourier series of functions of bounded variation. We also consider absolutely continuous functions and show by examples that a number of the results we obtain are "best possible".