Cesàro Means of Subsequences of Partial Sums of Trigonometric Fourier Series

Abstract

In 1936 Zygmunt Zalcwasser asked, with respect to the trigonometric system, how “rare” can a sequence of strictly monotone increasing integers \((n_j)\) be such that the almost everywhere relation \(\frac{1}{N}\sum _{j=1}^N S_{n_j}f \rightarrow f\) 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.