Abstract
Letf be a continuous function on the unit circle Γ, whose Fourier series is ω-absolutely convergent for some weight ω on the set of integersZ. If f is nowhere vanishing on Γ, then there exists a weightv onZ such that 1/f hadv-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 off, then there exists a weight Χ on Z such that φ ◯f has Χ-absolutely convergent Fourier series. This is a weighted analogue of Lévy’s generalization of Wiener’s theorem. In the theorems,v and Χ are non-constant if and only if ω is non-constant. In general, the results fail ifv or Χ is required to be the same weight ω.
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References
Edwards R-E, Fourier Series, (New York: Holt, Rinehart and Winston Inc.) (1967) vol.II
Gelfand I, Raikov D and Shilov G, Commutative normed rings (New York: Chelse Publication Company) (1964)
Reiter H and Stegeman J D, Classical harmonic analysis and locally compact abelian groups (Oxford: Clarendon Press) (2000)
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Bhatt, S.J., Dedania, H.V. Beurling algebra analogues of the classical theorems of Wiener and Lévy on absolutely convergent fourier series. Proc. Indian Acad. Sci. (Math. Sci.) 113, 179–182 (2003). https://doi.org/10.1007/BF02829767
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DOI: https://doi.org/10.1007/BF02829767