de la Vall\'ee Poussin Means of Walsh-Fourier Expansions

TL;DR

This paper investigates the convergence properties of de la Vallée Poussin means in Walsh-Fourier series, establishing criteria for convergence and divergence across various function classes.

Contribution

It provides a sharp criterion for almost everywhere convergence and constructs divergence examples in certain Orlicz classes.

Findings

Established a sharp convergence criterion for integrable functions.

Showed divergence in specific Orlicz classes when the criterion fails.

Linked convergence behavior to the structure of function classes.

Abstract

We study de la Vall\'ee Poussin means of Walsh--Fourier series associated with a nondecreasing window sequence. We establish a sharp criterion for almost everywhere convergence for integrable functions. We further show that, when this criterion fails, every Orlicz class below the logarithmic square-root scale contains a function whose de la Vall\'ee Poussin means diverge everywhere.