关于绝对收敛傅里叶级数的Wiener和Levy经典定理的Beurling代数类似
Beurling algebra analogues of the classical theorems of Wiener and Levy on absolutely convergent Fourier series
摘要：
设$f$是单位圆$Γ$上的连续函数，其Fourier级数对整数集$\mathcal{Z}$上的某个权$ω$是$ω$-绝对收敛的.若$f$在$Γ$上不为零，则存在$\mathcal{Z}$上的权$ν$，使得$1/f$具有$ν$-绝对收敛的傅里叶级数.这包括维纳的经典定理。作为推论，如果$ϕ$在$f$的值域的邻域上是全纯的，则存在$\mathcal{Z}$上的权$χ$，使得\hbox{$ϕ\circ f$}具有$χ$-绝对收敛的傅里叶级数.这是一个加权模拟列维的推广维纳定理。在定理中，$ν$和$χ$是非常数的当且仅当$ω$是非常数的。通常，如果要求$ν$或$χ$具有相同的权重$ω$，则结果失败。
英文摘要：
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 $ω$.
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