  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.
