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We present some simple counterexamples, based on quadratic forms ininfinitely many variables, showing that the implication $f\inC^{(\infty}(\mathbb{T}^\omega)\Longrightarrow\sum_{\bar{p}\in\mathbb{Z}^\infty}|\widehat{f}(\bar{p})|<\infty$is false. There are functions of the class $C^{(\infty}(\mathbb{T}^\omega)$(depending on an infinite number of variables) whose Fourier series divergesabsolutely. This fact establishes a significant difference from the finitedimensional case.

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