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

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 $\ell^p(ω)$, for some weight $ω$. When $ω=\{(k+1)^α\}_{k\in \mathbb{N}}$, for a fixed $α\in \mathbb{R}$, 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.