Almost everywhere convergence of sequences of two-dimensional Walsh-Fejér means of integrable functions

Abstract

The aim of this paper is to prove the a.e. convergence of sequences of the Fej\'er means of the Walsh-Fourier series of two variable integrable functions. That is, let $a = (a_1, a_2): \mathbb N \to \mathbb N^2$ such that $a_j(n+1)\ge\delta \sup_{k\le n}a_j(n)$ $(j=1,2, \, n\Ne)$ for some $\delta>0$ and $a_1(+\infty)=a_2(+\infty)=+\infty$. Then for each integrable function $f\in L^1(I^2)$ we have the a.e. relation $\lim_{n\to\infty}\sigma_{a_1(n), a_2(n)}f = f$. It will be a straightforward and easy consequence of this result the cone restricted a.e. convergence of the two-dimensional Walsh-Fej\'er means of integrable functions which was proved earlier by the author and Weisz \cite{gat2dim, w}.