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

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. 179-182. ISSN 0253-4142

[img]
Preview
PDF - Publisher Version
38kB

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 ω-absolutely 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 υ-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 ƒ, then there exists a weight χ on Z such that φ Ο ƒ has χ-absolutely convergent Fourier series. This is a weighted analogue of Lèvy's generalization of Wiener's theorem. In the theorems, υ and χ are non-constant if and only if ω is non-constant. 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

Repository Staff Only: item control page