Beurling Algebra Analogues of the Classical Theorems of Wiener and Lévy on Absolutely Convergent Fourier Series

Proceedings Mathematical Sciences

H V Dedania¹  S J Bhatt² 

[1] $$;Department of Mathematics, Sardar Patel University,VallabhVidyanagar 0, India$$

关键词: Fourier series;    Wiener’s theorem;    Lévy’s theorem;    Beurling algebra;    commutative Banach algebra.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 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 $varphicirc f$ 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.

【 授权许可】

Unknown