On model spaces and density of functions smooth on the boundary

TL;DR

This paper characterizes when functions with smooth boundary extensions are dense in model spaces, linking this property to the concentration of associated singular measures on specific sets, and provides a constructive method for approximation.

Contribution

It establishes a precise criterion for density of smooth boundary functions in model spaces based on measure concentration, using a duality argument and constructive approximation methods.

Findings

Density of smooth boundary functions occurs iff the singular measure is concentrated on countable unions of Beurling-Carleson sets.

If the measure's restriction avoids positive measure on Beurling-Carleson sets, larger function classes are not dense.

A constructive approach is used to demonstrate the existence of smooth approximants, contrasting earlier non-constructive results.

Abstract

We characterize the model spaces $K_\Theta$ in which functions with smooth boundary extensions are dense. It is shown that such approximations are possible if and only if the singular measure associated to the singular inner factor of $\Theta$ is concentrated on a countable union of Beurling-Carleson sets. In fact, we use a duality argument to show that if there exists a restriction of the associated singular measure which does not assign positive measure to Beurling-Carleson sets, then even larger classes of functions, such as H\"older classes and large collections of analytic Sobolev spaces, fail to be dense. In contrast to earlier results on density of functions with continuous extensions to the boundary in $K_\Theta$ and related spaces, the existence of a smooth approximant is obtained through a constructive method.