  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.
