• Beurling algebra analogues of the classical theorems of Wiener and Lévy on absolutely convergent fourier series

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      https://www.ias.ac.in/article/fulltext/pmsc/113/02/0179-0182

    • Keywords

       

      Fourier series; Wiener’s theorem; Lévy’s theorem; Beurling algebra; commutative Banach algebra

    • Abstract

       

      Letf be a continuous function on the unit circle Γ, whose Fourier series is ω-absolutely convergent for some weight ω on the set of integersZ. If f is nowhere vanishing on Γ, then there exists a weightv onZ such that 1/f hadv-absolutely convergent Fourier series. This includes Wiener’s classical theorem. As a corollary, it follows that if φ is holomorphic on a neighbourhood of the range off, then there exists a weight Χ on Z such that φ ◯f has Χ-absolutely convergent Fourier series. This is a weighted analogue of Lévy’s generalization of Wiener’s theorem. In the theorems,v and Χ are non-constant if and only if ω is non-constant. In general, the results fail ifv or Χ is required to be the same weight ω.

    • Author Affiliations

       

      S J Bhatt1 H V Dedania1

      1. Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar - 388 120, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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