Almost everywhere convergence of subsequences of Walsh-N\"orlund means

Abstract

Let $\{q_{k}: k\in\mathbb{N}\}$ be a non-increasing and convex sequence of non-negative numbers such that $q_{0}>0$. Our main result is to prove almost everywhere convergence \[ t_{a_{n}}(f)\to f \] of $f\in L_{1}(G)$ as $n\to\infty$, using several $\{a_{n}: n\in\mathbb{P}\}$ subsequences of positive integers for N\"orlund means (defined by sequence $q$) on the Walsh-Paley system with weaker conditions than it was known before.