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arXiv · 1906.02300

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

E. Fernández·L. Roncal
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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.

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E. Fernández, L. Roncal. 2019-06-05. On the absolute divergence of Fourier series in the infinite dimensional torus. https://arxiv.org/abs/1906.02300

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