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

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