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

Pub Date : 2024-03-06 DOI:10.1134/s0081543823050097

S. V. Konyagin

引用次数: 0

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.

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论多重三角傅里叶级数部分和的后继普林塞姆收敛性

Abstract A. N. Kolmogorov 1925 年的著名定理意味着，对于所有的 \(p\in(0,1)\) ，一个变量的任何可积分函数 \(f\) 的傅里叶级数的部分和都会收敛到 \(L^p\) 中。众所周知，这对于多变量函数来说并不成立。在本文中，我们证明了，尽管如此，对于任何几个变量的函数，都存在一个普林塞姆偏和子序列，对于所有的（p\in(0,1)\），这个子序列都收敛到了\(L^p\)中的函数。与此同时，在一种相当普遍的情况下，当我们求几个变量的函数在一个扩展的索引集系统上的傅里叶级数的偏和时，存在这样一个函数，对它来说，这些偏和的某个子序列的绝对值几乎在所有地方都趋向于无穷大。对于固定有界凸体的扩张系统和双曲交叉来说，尤其如此。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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