Convergence of sequences of two-dimensional Fejér means of trigonometric Fourier series 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 trigonometric Fourier series of two variable integrable functions. That is, let $a = (a_1, a_2): \mathbb N \to \mathbb N^2$ such that $a_j(n+1)\ge\alpha \sup_{k\le n}a_j(k)$ $(j=1,2, \, n\Ne)$ for some $\alpha>0$ and $a_1(+\infty)=a_2(+\infty)=+\infty$. Then for each integrable function $f\in L^1(\mathbb{T}^2)$ we have the a.e. relation $\lim_{n\to\infty}\sigma_{a(n)}f = f$. It will be a straightforward and easy consequence of this result the historical cone restricted a.e. convergence result with respect to the two-dimensional Fej\'er means of integrable functions due to Marcinkiewicz and Zygmund \cite{mz}.