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

Abstract

Let $f$ 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 $\mathcal{Z}$. If $f$ is nowhere vanishing on $Γ$, then there exists a weight $ν$ on $\mathcal{Z}$ such that $1/f$ had $ν$-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 $f$, then there exists a weight $χ$ on $\mathcal{Z}$ such that \hbox{$ϕ\circ f$} has $χ$-absolutely convergent Fourier series. This is a weighted analogue of Lévy's generalization of Wiener's theorem. In the theorems, $ν$ and $χ$ are non-constant if and only if $ω$ is non-constant. In general, the results fail if $ν$ or $χ$ is required to be the same weight $ω$.