2003/10/18 by S. J. Bhatt, H. V. Dedania · 3 citations
Mathematics · #math.CV #math.CA
published as Proc. Indian Acad. Sci. (Math. Sci.), Vol. 113, No. 2, May 2003, pp. 179-182 · 4 pages
arxiv created 2003/10/18 · arxiv updated 2009/12/01
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 Z. If f is nowhere vanishing on Γ, then there exists a weight ν on 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 Z such that \hboxϕ∘ 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.