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

Abstract

In this note we present some simple counterexamples, based on quadratic forms in infinitely many variables, showing that the implication $f\in C^{(\infty}(\mathbb{T}^\omega)\Longrightarrow\sum_{\bar{p}\in\mathbb{Z}^\infty}|\widehat{f}(\bar{p})|<\infty$ 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 to what happens in the finite dimensional case.BCAM Severo Ochoa excellence accreditation SEV-2013-0323 MTM2015-65888-C04-4-P 2017 Leonardo grant for Researchers and Cultural Creators, BBVA Foundatio