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

Abstract

We study the cyclic vectors and the spanning set of the circle for the p β (Z) spaces of all sequences u = u n n∈Z such that u n (1 + |n|) β n∈Z ∈ p (Z) with p > 1 and β > 0. By duality the spanning set is the uniqueness set of the distribution on the circle whose Fourier coefficients are in q −β (Z) where q is the conjugate of p. Our characterizations are given in terms of the Hausdorff dimension and capacity.