Cyclicity in reproducing kernel Hilbert spaces of analytic functions

We introduce a large family of reproducing kernel Hilbert spaces $$\mathcal{H}$ \subset \mbox{Hol}($\mathbb{D}$)$, which include the classical Dirichlet-type spaces ${\mathcal{D}}_{\alpha }$, by requiring normalized monomials to form a Riesz basis for $\mathcal{H}$. Then, after precisely evaluating the $n$-th optimal norm and the $n$-th approximant of $f(z)=1-z$, we completely characterize the cyclicity of functions in $Hol(\bar{\mathbb{D}})$ with respect to the forward shift.