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Beurling algebra analogues of the classical theorems of Wiener and Levy on absolutely convergent Fourier series | | S. J. Bhatt
; H. V. Dedania
; | | Date: |
18 Oct 2003 | | Journal: | Proc. Indian Acad. Sci. (Math. Sci.), Vol. 113, No. 2, May 2003, pp. 179-182 | | Subject: | Complex Variables; Classical Analysis and ODEs | math.CV math.CA | | Abstract: | Let $f$ be a continuous function on the unit circle $Gamma$, whose Fourier series is $omega$-absolutely convergent for some weight $omega$ on the set of integers $mathcal{Z}$. If $f$ is nowhere vanishing on $Gamma$, then there exists a weight $
u$ on $mathcal{Z}$ such that $1/f$ had $
u$-absolutely convergent Fourier series. This includes Wiener’s classical theorem. As a corollary, it follows that if $phi$ is holomorphic on a neighbourhood of the range of $f$, then there exists a weight $chi$ on $mathcal{Z}$ such that hbox{$phicirc f$} has $chi$-absolutely convergent Fourier series. This is a weighted analogue of Lévy’s generalization of Wiener’s theorem. In the theorems, $
u$ and $chi$ are non-constant if and only if $omega$ is non-constant. In general, the results fail if $
u$ or $chi$ is required to be the same weight $omega$. | | Source: | arXiv, math.CV/0310291 | | Services: | Forum | Review | PDF | Favorites | |
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