Constructive approximation in de Branges-Rovnyak spaces

TL;DR

This paper investigates approximation properties in de Branges-Rovnyak spaces, showing that dilations do not always approximate functions in norm, and provides a constructive proof of polynomial density in these spaces.

Contribution

It demonstrates that dilation-based approximation fails in certain de Branges-Rovnyak spaces and offers the first constructive proof of polynomial density in these spaces.

Findings

Dilation approximation can diverge in de Branges-Rovnyak spaces.

Polynomials are dense in these spaces for non-extreme points of the unit ball.

Constructive proof of polynomial density is provided.

Abstract

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 $f\_r(z):=f(rz)~(r \textless{} 1)$. We show that this is \emph{not} the case for the de Branges--Rovnyak spaces $\cH(b)$. More precisely, we give an example of a non-extreme point $b$ of the unit ball of $H^\infty$ and a function $f\in\cH(b)$ such that $\lim\_{r\to1^-}\|f\_r\|\_{\cH(b)}=\infty$. It is known that, if $b$ is a non-extreme point of the unit ball of $H^\infty$, then polynomials are dense in $\cH(b)$. We give the first constructive proof of this fact.