Beurling algebra analogues of the classical theorems of Wiener and Lèvy on absolutely convergent Fourier series

Abstract

Let ƒ be a continuous function on the unit circle Γ, whose Fourier series is ω-absolutely convergent for some weight ω on the set of integers Z. If f is nowhere vanishing on Γ, then there exists a weightv on Z such that 1/ƒ had υ-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 of ƒ, then there exists a weight χ on Z such that φ Ο ƒ has χ-absolutely convergent Fourier series. This is a weighted analogue of Lèvy's generalization of Wiener's theorem. In the theorems, υ and χ are non-constant if and only if ω is non-constant. In general, the results fail if υ or ƒ is required to be the same weight ω.