A note on Vilenkin-Fej\'er means on the Martingale Hardy spaces $H_{p}$

TL;DR

This paper establishes necessary and sufficient conditions for the convergence of Fejér means in Martingale Hardy spaces $H_p$ using the modulus of continuity, enhancing understanding of harmonic analysis in these spaces.

Contribution

It provides new criteria linking Fejér means convergence to the modulus of continuity in Hardy spaces $H_p$, for $0<p extless=1$, which was previously unexplored.

Findings

Derived necessary and sufficient conditions for convergence

Connected Fejér means convergence to modulus of continuity

Extended analysis to Hardy spaces with $0<p extless=1$

Abstract

The main aim of this note is to derive necessary and sufficient conditions for the convergence of Fej\'er means in terms of the modulus of continuity of the Hardy spaces $H_{p},$ $\left(0<p\leq 1\right)$.