Almost Everywhere Convergence of Subsequence of Quadratic Partial Sums of Two-Dimensional Walsh–Fourier Series

Abstract

For a non-negative integer n let us denote the dyadic variation of a natural number n by

$$V(n): = \sum\limits_{j = 0}^\infty {\left| {{n_j} - {n_{j + 1}}} \right| + {n_0},}$$

where n := ∑ ^∞_i=0 n _i 2ⁱ, n _i ∈ {0, 1}. In this paper we prove that for a function f ∈ L log L(I²) under the condition sup _A V (nA) < ∞, the subsequence of quadratic partial sums $S{_n^{\square}}_A\left( f \right)$ of two-dimensional Walsh–Fourier series converges to the function f almost everywhere. We also prove sharpness of this result. Namely, we prove that for all monotone increasing function φ: [0,∞) → [0,∞) such that φ(u) = o(u log u) as u → ∞ there exists a sequence {n _A : A ≥ 1} with the condition sup _A V(nA) < ∞ and a function f ∈ φ(L)(I²) for which $\text{sup} _A|S{_n^{\square}}_A\left( {{x^1},{x^2};f} \right)| = \infty$ for almost all (x¹, x²) ∈ I².