Sequences of partial sums of multiple trigonometric Fourier series

Abstract

Let $f$ be an integrable $2\pi$-periodic function of $d\ge2$ variables. For a bounded subset $A$ of the $d$-dimensional space let $S_A(f)$ denote the sum of terms of the Fourier series of $f$ with frequencies in $A$. The following problem is addressed: given a sequence $\{A_j\}$ of bounded convex sets, do there exist a function $f$ and a sequence $\{j_\nu\}$ such that $\lim_{\nu\to\infty} |S_{A_{j_\nu}} (f)|=\infty$ almost everywhere? Bibliography: 5 titles.