Polynomial approach to cyclicity for weighted $\ell ^p_A$

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 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.