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 Fejer means of the Walsh–Fourier series of bivariate integrable functions. That is, let \(a = (a_{1}, a_{2})\:\mathbb{N} \to \mathbb{N}^{2}\) such that a j (n+1)≧δsup k≦n a j (n) (j=1,2, n∈ℕ) for some δ>0 and a 1(+∞)=a 2(+∞)=+∞. Then for each integrable function f∈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–Fejer means of integrable functions which was proved earlier by the author and Weisz [3,8].