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

S. J. Bhatt¹ ; P. A. Dabhi¹ ; H. V. Dedania¹

¹ Department of Mathematics Sardar Patel University Vallabh Vidyanagar-388 120 Gujarat, India

Studia Mathematica, Tome 195 (2009) no. 3, pp. 219-225

Résumé

Let $0 p\leq 1$, let $\omega :\mathbb Z\rightarrow [1,\infty)$ be a weight on $\mathbb{Z}$ and let $f$ be a nowhere vanishing continuous function on the unit circle $\mit\Gamma$ whose Fourier series satisfies $\sum _{n\in \mathbb{Z}}{|\widehat f(n)|^p \omega(n)} \infty$. Then there exists a weight $\nu$ on $\mathbb{Z}$ such that $\sum _{n\in \mathbb{Z}}{|\widehat{(1/f)}(n)|^p \nu(n)} \infty$. Further, $\nu$ is non-constant if and only if $\omega$ is non-constant; and $\nu=\omega$ if $\omega$ is non-quasianalytic. This includes the classical Wiener theorem ($p=1$, $\omega= 1$), Domar theorem ($p=1$, $\omega$ is non-quasianalytic), Żelazko theorem ($\omega=1$) 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 $\omega$ and $0 p 1$, the locally bounded space $L^p(G,\omega)$ 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

DOI : 10.4064/sm195-3-2

Mots-clés : leq omega mathbb rightarrow infty weight mathbb nowhere vanishing continuous function unit circle mit gamma whose fourier series satisfies sum mathbb widehat omega infty there exists weight mathbb sum mathbb widehat infty further non constant only omega non constant omega omega non quasianalytic includes classical wiener theorem omega domar theorem omega non quasianalytic elazko theorem omega recent result bhatt dedania analogue theorem present level generality developed given locally compact group continuous weight omega locally bounded space omega closed under convolution only discrete only admits atom generalizes refines another result elazko