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

Abstract

We study the cyclic vectors and the spanning set of the circle for the ${\ell }_{\beta }^p(\mathbb{Z})$ 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 ${\ell }_{-\beta }^q(\mathbb{Z})$ is the spanning set for the ${\ell }_{\beta }^p(\mathbb{Z})$ spaces, where $q$ is the conjugate of $p$. Our characterizations are given in terms of the Hausdorff dimension and capacity.