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

## TL;DR

This paper demonstrates that in infinite-dimensional tori, smooth functions can have Fourier series that diverge absolutely, contrasting with the finite-dimensional case where smoothness implies absolute convergence.

## Contribution

It provides simple counterexamples showing that smoothness does not guarantee absolute convergence of Fourier series in infinite dimensions.

## Key findings

- Counterexamples of divergence for smooth functions
- Difference between finite and infinite-dimensional Fourier series behavior
- Establishment of divergence despite smoothness in infinite dimensions

## Abstract

We present some simple counterexamples, based on quadratic forms in infinitely many variables, showing that the implication $f\in C^{(\infty}(\mathbb{T}^\omega)\Longrightarrow\sum_{\bar{p}\in\mathbb{Z}^\infty}|\widehat{f}(\bar{p})|<\infty$ 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 from the finite dimensional case.