Almost Everywhere Convergence of Arithmetic Means of Walsh--Fourier Partial Sums Along Subsequences

TL;DR

This paper extends the almost everywhere convergence of arithmetic means of Walsh-Fourier partial sums along subsequences to a broader range of growth conditions, specifically for all 0<δ<1.

Contribution

It generalizes Gát's 2019 result by proving convergence for the entire range 0<δ<1, broadening the applicability of Walsh-Fourier series convergence along subsequences.

Findings

Convergence holds for all 0<δ<1, expanding previous results.

The growth condition on the subsequence is less restrictive than before.

Almost everywhere convergence is established for a wider class of subsequences.

Abstract

Let $S_m f$ denote the $m$-th partial sum of the Walsh-Fourier series of $f \in L^1$. For an increasing sequence $a=(a(n))_{n \geq 1}$ of positive integers, consider the arithmetic means $$ \sigma_N f:=\frac{1}{N} \sum_{n=1}^N S_{a(n)} f . $$ G\'at proved in 2019 that $\sigma_N f \rightarrow f$ almost everywhere for every $f \in L^1$ under the growth condition $$ a(n+1) \geq\left(1+\frac{1}{n^\delta}\right) a(n), \quad 0<\delta<\frac{1}{2} . $$ We show that the same conclusion remains valid throughout the full range $0<\delta<1$.