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

• # Fulltext

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

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

• # Proceedings – Mathematical Sciences

Volume 132, 2022
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019