On the partial sums of Walsh-Fourier series

TL;DR

This paper studies the convergence and divergence behavior of specific subsequences of Walsh-Fourier partial sums on martingale Hardy spaces, linking convergence ratios to the modulus of continuity, providing necessary and sufficient conditions.

Contribution

It introduces new conditions relating Walsh-Fourier partial sums convergence to the modulus of continuity in martingale Hardy spaces, establishing their necessity and sufficiency.

Findings

Identifies conditions for convergence of Walsh-Fourier partial sums.

Establishes divergence criteria for certain subsequences.

Links convergence behavior to the modulus of continuity.

Abstract

In this paper we investigate some convergence and divergence of some specific subsequences of partial sums with respect to Walsh system on the martingale Hardy spaces. By using these results we obtain relationship of the ratio of convergence of the partial sums of the Walsh series with the modulus of continuity of martingale. These conditions are in a sense necessary and sufficient.