On Pringsheim Convergence of a Subsequence of Partial Sums of a Multiple Trigonometric Fourier Series

Abstract

A. N. Kolmogorov’s famous theorem of 1925 implies that the partial sums of the Fourier series of any integrable function \(f\) of one variable converge to it in \(L^p\) for all \(p\in(0,1)\). It is known that this does not hold true for functions of several variables. In this paper we prove that, nevertheless, for any function of several variables there is a subsequence of Pringsheim partial sums that converges to the function in \(L^p\) for all \(p\in(0,1)\). At the same time, in a fairly general case, when we take the partial sums of the Fourier series of a function of several variables over an expanding system of index sets, there exists a function for which the absolute values of a certain subsequence of these partial sums tend to infinity almost everywhere. This is so, in particular, for a system of dilations of a fixed bounded convex body and for hyperbolic crosses.