Almost Everywhere Convergence of Arithmetic Means of Walsh--Fourier Partial Sums Along 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 $a=(a(n){)}_{n\ge 1}$ of positive integers, consider the arithmetic means $${\sigma }_{N}f:=\frac{1}{N}\sum _{n=1}^N{S}_{a(n)}f.$$ Gát proved in 2019 that ${\sigma }_{N}f\to f$ almost everywhere for every $f\in L^1$ under the growth condition $$a(n+1)\ge \left(1+\frac{1}{n^{\delta }}\right)a(n),0<\delta <\frac{1}{2}.$$ We show that the same conclusion remains valid throughout the full range $0<\delta <1$.