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

Let $f$ be a continuous function on the unit circle $\mathrm{\Gamma }$, whose Fourier series is $\omega$-absolutely convergent for some weight $\omega$ on the set of integers $\mathcal{Z}$. If $f$ is nowhere vanishing on $\mathrm{\Gamma }$, then there exists a weight $\nu$ on $\mathcal{Z}$ such that $1\mathrm{/}f$ had $\nu$-absolutely convergent Fourier series. This includes Wiener’s classical theorem. As a corollary, it follows that if $\varphi$ is holomorphic on a neighbourhood of the range of $f$, then there exists a weight $\chi$ on $\mathcal{Z}$ such that \hbox{$\varphi \circ f$} has $\chi$-absolutely convergent Fourier series. This is a weighted analogue of L'{e}vy’s generalization of Wiener’s theorem. In the theorems, $\nu$ and $\chi$ are non-constant if and only if $\omega$ is non-constant. In general, the results fail if $\nu$ or $\chi$ is required to be the same weight $\omega$.