Subsequences of spherical sums of double Walsh-Fourier series

Abstract

In this paper we prove the following: let $1\leq p<2,$ then the set of the functions from the space $L_{p}(\mathbb{I}^{2})$ with a subsequence of spherical partial sums of the double Walsh-Fourier series convergent in
measure on $\mathbb{I}^{2}$ is of first Baire category in $L_{p}(\mathbb{I}% ^{2}).$ We also prove that for each function $f\in L_{2}(\mathbb{I}^{2})$ a.e. convergence $S_{R({n})}^{\circ }\left( f\right) \rightarrow f$ holds,
where $R(n)$ is a lacunary sequence of positive integers.