Limits of sequences of operators associated with Walsh System

TL;DR

This paper establishes the exact conditions on weights for the almost everywhere convergence of Walsh system operators and examines their behavior in tensor product cases, highlighting limitations shown in recent research.

Contribution

It provides necessary and sufficient conditions for weights to ensure convergence of Walsh operators, extending understanding of their limits and behavior in tensor product scenarios.

Findings

Identifies conditions for almost everywhere convergence of Walsh operators.

Shows tensor product operators generally do not converge almost everywhere.

Provides criteria for convergence in measure on L1 space.

Abstract

The aim of the current paper is to determine the necessary and sufficient conditions for the weights $\mathbf{q}=\{q_k\}$, ensuring that the sequence of operators $\left\{ T_{n}^{\left( \mathbf{q}\right) }f\right\} $ associated with Walsh system, is convergent almost everywhere for all integrable function $f$. The article also examines the convergence of a sequence of tensor product operators denoted as $\left\{ T_{n}^{( \mathbf{q})}\otimes T_{n}^{( \mathbf{p})}\right\}$ involving functions of two variables. We point out that recent research by G\'{a}t and Karagulyan (2016) demonstrated that this sequence of tensor product operators cannot converge almost everywhere for every integrable function. In this paper, the necessary and sufficient conditions for the weight are provided which ensure that the sequence of the mentioned operators converges in measure on $L_{1}$.