Convergence of a Subsequence of Triangular Partial Sums of Double Walsh-Fourier Series

Abstract

In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ L_p such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space L_p(II²) (1 ≤ p < 2) with subsequence of triangular partial means \(S_{2^A}^\Delta(f)\) of the double Walsh-Fourier series convergent in measure on II² is of first Baire category in L_p(II²). We also prove that for each function f ∈ L₂(II²) a.e. convergence \(S_{a(n)}^\Delta (f) \rightarrow f\) holds, where a(n) is a lacunary sequence of positive integers.