Constructive approximation in de Branges-Rovnyak spaces

In most classical holomorphic function spaces on the unit disk, a function $f$ can be approximated in the norm of the space by its dilates $f\mathrm{\_}r(z):=f(rz)(r\text{textless}1)$. We show that this is \emph{not} the case for the de Branges–Rovnyak spaces $\text{cH}(b)$. More precisely, we give an example of a non-extreme point $b$ of the unit ball of $H^{\mathrm{\infty }}$ and a function $f\in \text{cH}(b)$ such that $\lim \mathrm{\_}r\to 1^{-}\mathrm{\parallel }f\mathrm{\_}r\mathrm{\parallel }\mathrm{\_}\text{cH}(b)=\mathrm{\infty }$. It is known that, if $b$ is a non-extreme point of the unit ball of $H^{\mathrm{\infty }}$, then polynomials are dense in $\text{cH}(b)$. We give the first constructive proof of this fact.