Ces\`{a}ro means of subsequences of partial sums of trigonometric Fourier series

TL;DR

This paper investigates the conditions under which Cesàro means of subsequences of partial sums of trigonometric Fourier series converge almost everywhere, providing new insights into the behavior of lacunary sequences.

Contribution

It proves that Cesàro means of lacunary subsequences of Fourier partial sums converge almost everywhere for all integrable functions, answering a longstanding question.

Findings

Cesàro means of lacunary subsequences converge a.e. for all integrable functions.

The result applies to any lacunary sequence of natural numbers.

It extends the understanding of convergence of Fourier series.

Abstract

In 1936 Zygmunt Zalcwasser asked with respect to the trigonometric system that how "rare" can a sequence of strictly monotone increasing integers $(n_j)$ be such that the almost everywhere relation $\frac{1}{N}\sum_{j=1}^N S_{n_j}f \to f$ is fulfilled for each integrable function $f$. In this paper, we give an answer to this question. It follows from the main result that this a.e. relation holds for every integrable function $f$ and lacunary sequence $(n_j)$ of natural numbers.