Ces`aro means of negative order of the one-dimensional Vilenkin-Fourier series

Abstract

In [1] has been proved some inequalities related to the approximation properties of Ces`aro means of negative order of the one-dimensional Vilenkin-Fourier series. These inequalities allow one to obtain a sufficient condition for the convergence of Ces`aro means of VilenkinFourier series in the L^p− metric in the term of modulus of continuity. In this paper, we will prove the sharpness of these conditions, in particular we find a continuous function under some condition of modulo of continuity, for which Ces`aro means of Vilenkin-Fourier series diverge in the L^p− metric.