Polynomial approach to cyclicity for weighted ${\ell }_A^p$

Abstract

In previous works, an approach to the study of cyclic functions in reproducing kernel Hilbert spaces has been presented, based on the study of so called \emph{optimal polynomial approximants}. In the present article, we extend such approach to the (non-Hilbert) case of spaces of analytic functions whose Taylor coefficients are in ${\ell }^p(\omega )$, for some weight $\omega$. When $\omega =\{(k+1{)}^{\alpha }{\}}_{k\in \mathbb{N}}$, for a fixed $\alpha \in \mathbb{R}$, we derive a characterization of the cyclicity of polynomial functions and, when $1<p<\infty$, we obtain sharp rates of convergence of the optimal norms.

Major revision