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On the absolute divergence of Fourier series on the infinite-dimensional torus

Volume 157 / 2019

Emilio Fernández, Luz Roncal 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.

Authors

  • Emilio FernándezDepartamento de Matemáticas y Computación
    Universidad de La Rioja
    c/ Madre de Dios, 53
    26006 Logroño, Spain
    e-mail
  • Luz RoncalBCAM – Basque Center for
    Applied Mathematics
    Alameda de Mazarredo, 14
    E-48009 Bilbao, Basque Country, Spain
    e-mail

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