On the absolute divergence of Fourier series in the infinite dimensional torus

Abstract

We present some simple counterexamples, based on quadratic forms in infinitely many variables, showing that the implication $f\in C^{(\infty }({\mathbb{T}}^{\omega })⟹{\sum }_{\bar{p}\in {\mathbb{Z}}^{\infty }}|\hat{f}(\bar{p})|<\infty$ is false. There are functions of the class $C^{(\infty }({\mathbb{T}}^{\omega })$ (depending on an infinite number of variables) whose Fourier series diverges absolutely. This fact establishes a significant difference from the finite dimensional case.

12 pages