# Outer functions and divergence in de Branges-Rovnyak spaces

## TL;DR

The paper constructs a de Branges-Rovnyak space with dense polynomials where a function's dilates have unbounded norm, highlighting unique divergence behavior linked to the outer function property of b.

## Contribution

It demonstrates the existence of a de Branges-Rovnyak space with dense polynomials and an outer function b where dilates of a specific function diverge in norm.

## Key findings

- Polynomials are dense in the constructed space.
- A function with diverging dilate norms exists in this space.
- The divergence is tied to the outer nature of b.

## Abstract

In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function $f$ can be approximated in norm by its dilates $f_r(z):=f(rz)~(r<1)$, in other words, $\lim_{r\to1^-}\|f_r-f\|=0$. We construct a de Branges-Rovnyak space ${\mathcal H}(b)$ in which the polynomials are dense, and a function $f\in{\mathcal H}(b)$ such that $\lim_{r\to1^-}\|f_r\|_{{\mathcal H}(b)}=\infty$. The essential feature of our construction lies in the fact that $b$ is an outer function.