On the Approximation Properties of Ces\`{a}ro Means of Negative Order of Vilenkin-Fourier Series

Abstract

In this paper we establish approximation properties of Ces\`{a}ro $(C,-\alpha )$ means with $\alpha$ $\epsilon$ $(0,1)$ of Vilenkin-Fourier series.This result allows one to obtain a condition which is sufficient for the convergence of the means $\sigma_n^{-\alpha }(f,x)$ to $f(x)$ in the $L^p-$metric.