subsequences of Triangular Partial Sums of Double Fourier Series on Unbounded Vilenkin Groups

Abstract

In 1987 Harris proved-among others-that for each $1\leq p<2$ there exists a
two-dimensional function $f\in L_{p}$ such that its triangular partial sums $%
S_{2^{A}}^{\triangle }f$ of Walsh-Fourier series does not converge almost
everywhere. In this paper we prove that subsequences of triangular partial
sums $S_{n_{A}M_{A}}^{\triangle }f,n_{A}\in \left\{ 1,2,...,m_{A}-1\right\} $
on unbounded Vilenkin groups converge almost everywhere to $f$ for each
function $f\in L_{2}.$