ON THE RATE OF CONVERGENCE OF CESARO MEANS OF WALSH-FOURIER SERIES

The aim of this paper is to make a complete investigation concerning the interaction between the rate of convergence of Cesàro means of Walsh-Fourier series and the modulus of continuity. We give the best possible sufficient conditions with respect to the modulus of continuity that implies the convergence at a given rate. We also give the best necessary conditions. These questions are studied in Lp (1 ≤ p < ∞) and in uniform norms. As a consequence, we receive the best results for the Lipschitz classes. The solution of a problem of F. Moó ricz and A. H. Siddiqi (1992, J. Approx. Theory70, 375-389), i.e., the characterization of the Favard (saturation) classes of the Cesàro summation, can be derived from our theorems. © 1994 Academic Press, Inc.