Beurling algebra analogues of theorems of Wiener-Lévy-Żelazko and Żelazko

S. J. Bhatt ; P. A. Dabhi ; H. V. Dedania

Studia Mathematica, Tome 192 (2009), p. 219-225 / Harvested from The Polish Digital Mathematics Library

Résumé

Let 0 < p ≤ 1, let ω: ℤ → [1,∞) be a weight on ℤ and let f be a nowhere vanishing continuous function on the unit circle Γ whose Fourier series satisfies $∑_{n∈ℤ} |f̂(n)|^{p}ω(n) < ∞$. Then there exists a weight ν on ℤ such that $∑_{n∈ℤ} |\widehat{(1/f)}(n)|^{p} ν(n) < ∞$. Further, ν is non-constant if and only if ω is non-constant; and ν = ω if ω is non-quasianalytic. This includes the classical Wiener theorem (p = 1, ω = 1), Domar theorem (p = 1, ω is non-quasianalytic), Żelazko theorem (ω = 1) and a recent result of Bhatt and Dedania (p = 1). An analogue of the Lévy theorem at the present level of generality is also developed. Given a locally compact group G with a continuous weight ω and 0 < p < 1, the locally bounded space $L^{p}(G,ω)$ is closed under convolution if and only if G is discrete if and only if G admits an atom. This generalizes and refines another result of Żelazko.

Détail

Publié le : 2009-01-01

Zbl 1178.42001

EUDML-ID : urn:eudml:doc:284428