On the convergence of Fej\'er means of Walsh-Fourier series in the space $H_{p}$

TL;DR

This paper investigates the conditions under which Fejér means of Walsh-Fourier series converge in the Hardy space $H_p$ for $0<p extless=1/2$, focusing on the role of the modulus of continuity of martingales.

Contribution

It establishes necessary and sufficient conditions for Fejér means convergence in $H_p$-norm based on the modulus of continuity for martingales in this range.

Findings

Identifies precise modulus of continuity conditions for convergence

Provides a characterization for $0<p extless=1/2$

Advances understanding of Walsh-Fourier series in Hardy spaces

Abstract

The main aim of this paper is to find the necessary and sufficient conditions for a modulus of continuity of a martingale $F\in H_{p},$ for which Fej\'er means convergence in $H_{p}$-norm, when $0<p\leq 1/2.$