Wiener Algebras and Trigonometric Series in a Coordinated Fashion

E. Liflyand, R. Trigub

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

Let W(R) be the Wiener Banach algebra of functions representable by the Fourier integrals of Lebesgue integrable functions. It is proved in the paper that, in particular, a trigonometric series ∑k=-∞∞ckeikt is the Fourier series of an integrable function if and only if there exists a ϕ∈ W(R) such that ϕ(k) = ck, k∈ Z. If f∈ W(R) , then the piecewise linear continuous function ℓf defined by ℓf(k) = f(k) , k∈ Z, belongs to W(R) as well. Moreover, ‖ℓf‖W0≤‖f‖W0. Similar relations are established for more advanced Wiener algebras. These results are supplemented by numerous applications. In particular, new necessary and sufficient conditions are proved for a trigonometric series to be a Fourier series and new properties of W are established.

Original languageEnglish
Pages (from-to)185-206
Number of pages22
JournalConstructive Approximation
Volume54
Issue number2
DOIs
StatePublished - Oct 2021

Bibliographical note

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.

Keywords

  • Bernstein inequality
  • Fourier series of a measure
  • Fourier transform of a measure
  • Hilbert transform
  • Poisson summation formula
  • Wiener algebras, entire function of exponential type

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