Almost everywhere divergence of Cesaro means of subsequences of Walsh--Paley Fourier partial sums

Abstract

We prove an almost everywhere divergence theorem for Cesàro means of subsequences of partial sums of Walsh--Fourier series. More precisely, we show that there exists a strictly increasing sequence of positive integers $(a_n)$ and a function $f\in L^1(G)$, where $G$ denotes the dyadic group, such that \[ \left( \frac1N\sum_{n=1}^N S_{a_n}f(x) \right)_{N=1}^{\infty} \] does not converge for almost every $x\in G$. The result is motivated by the corresponding trigonometric problem. A problem going back to Zalcwasser (1936) and remaining open for almost ninety years was recently solved in the trigonometric setting by Gát, who proved almost everywhere divergence for suitable subsequential arithmetic means of partial sums. In the Walsh--Paley setting the situation is more delicate. The dyadic structure gives positive convergence results; for instance, the partial sums $S_{2^m}f$ converge almost everywhere to $f$ for every $f\in L^1(G)$. Our theorem shows that, despite this stabilizing dyadic structure, suitably chosen arithmetic means of Walsh--Fourier partial sums may still diverge almost everywhere.