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

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.