Strong convergence theorem of Ces\`aro means with respect to the Walsh system

TL;DR

This paper proves that Cesàro means of Walsh-Fourier series are uniformly bounded in the martingale Hardy space for certain p-values, advancing understanding of convergence in harmonic analysis.

Contribution

It establishes the strong convergence theorem of Cesàro means with respect to the Walsh system in martingale Hardy spaces for the first time.

Findings

Cesàro means are uniformly bounded operators in H_p for 0<p<1/(1+α)

The result extends convergence properties of Walsh-Fourier series

Provides new bounds in martingale Hardy spaces

Abstract

We prove that Ces\`{a}ro means of one-dimensional Walsh-Fourier series are uniformly bounded operators in the martingale Hardy space $H_{p}$ for $% 0<p<1/\left( 1+\alpha \right).$