On the absolute divergence of Fourier series on the infinite-dimensional torus
Volume 157 / 2019
Colloquium Mathematicum 157 (2019), 143-155
MSC: Primary 42B05; Secondary 43A50, 46G99.
DOI: 10.4064/cm7568-6-2018
Published online: 22 March 2019
Abstract
We present some simple counterexamples, based on quadratic forms in infinitely many variables, showing that the implication $f\in C^{(\infty}(\mathbb{T}^\omega)\Rightarrow\sum_{\bar{p}\in\mathbb{Z}^\infty}|\widehat{f}(\bar{p})| \lt \infty$ is false: there are functions of class $C^{(\infty}(\mathbb{T}^\omega)$ (depending on an infinite number of variables) whose Fourier series diverges absolutely. This is a significant difference from the finite-dimensional case.
