Spans of translates in weighted $\ell^p$ spaces

Abstract

We study the cyclic vectors and the spanning set of the circle for the $\ell^p\_\beta(\mathbb{Z})$ spaces of all sequences $u= (u\_n ){n\in \mathbb{Z}}$ such that $(u\_n (1+|n|)^{\beta} ){n\in \mathbb{Z}} \in \ell^p(\mathbb{Z})$ , with $p>1$ and $\beta>0$. The uniqueness set of the distribution on the circle whose Fourier coefficients are in $\ell^q\_{-\beta}(\mathbb{Z})$ is the spanning set for the $\ell^p\_\beta(\mathbb{Z})$ spaces, where $q$ is the conjugate of $p$. Our characterizations are given in terms of the Hausdorff dimension and capacity.