Bhatt, S. J. ; Dedania, H. V. (2003) Beurling algebra analogues of the classical theorems of Wiener and Lèvy on absolutely convergent Fourier series Proceedings of the Indian Academy of Sciences  Mathematical Sciences, 113 (2). pp. 179182. ISSN 02534142

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Official URL: http://www.ias.ac.in/mathsci/vol113/may2003/Pm2055...
Related URL: http://dx.doi.org/10.1007/BF02829767
Abstract
Let ƒ be a continuous function on the unit circle Γ, whose Fourier series is ωabsolutely 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 υabsolutely 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 χabsolutely convergent Fourier series. This is a weighted analogue of Lèvy's generalization of Wiener's theorem. In the theorems, υ and χ are nonconstant if and only if ω is nonconstant. In general, the results fail if υ or ƒ is required to be the same weight ω.
Item Type:  Article 

Source:  Copyright of this article belongs to Indian Academy of Sciences. 
Keywords:  Fourier Series; Wiener's Theorem; Lévy's Theorem; Beurling Algebra; Commutative Banach Algebra 
ID Code:  59679 
Deposited On:  07 Sep 2011 05:18 
Last Modified:  18 May 2016 10:09 
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