Pointwise convergence of multiple Fourier series using different convergence notions
Convergence in Pringsheim's sense of a multiple series follows from its regular convergence, but the converse is not true in general. In this paper we study the relation between the convergence notions applied to multiple Fourier series. The following two theorems are typical, where \(T^ d=\{x\in R^ d:\) \(0\leq x_ j<2\pi\) for \(j=1,2,...,d\}\) denotes the d-dimensional torus. Theorem 1. If \(f\in L^ p(T^ 2)\) for some \(p>1\), then the regular convergence and the convergence in Pringsheim's sense of the double Fourier series of f are equivalent, up to a set of measure zero. The situation is sharply different from \(d=3\) on. Theorem 3. There exists a function \(g\in C(T^ 3)\) whose triple Fourier series converges in Pringsheim's sense on a set E of positive measure, but fails to converge regularly a.e. on E.
scientific article; zbMATH DE number 3964642