On an approximation of 2-dimensional Walsh–Fourier series in martingale Hardy spaces

Abstract

In this paper, we investigate convergence and divergence of partial sums with respect to the $2$-dimensional Walsh system on the martingale Hardy spaces. In particular, we find some conditions for the modulus of continuity which provide convergence of partial sums of Walsh–Fourier series. We also show that these conditions are in a sense necessary and sufficient.