Thin Sequences and Their Role in Model Spaces and Douglas Algebras

TL;DR

This paper investigates thin interpolating sequences in Hardy and model spaces, revealing that under certain conditions, their interpolation properties are equivalent, advancing understanding of function theory in these spaces.

Contribution

It establishes a connection between interpolation in Hardy spaces and model spaces for thin sequences, especially when the inner function tends to zero at the sequence points.

Findings

Interpolation properties in $H^2$ and $K_\Theta$ are equivalent under certain conditions.

Results relate to Carleson measures and the behavior of inner functions.

Provides new insights into the structure of model spaces and their relation to Hardy space interpolation.

Abstract

We study thin interpolating sequences $\{\lambda_n\}$ and their relationship to interpolation in the Hardy space $H^2$ and the model spaces $K_\Theta = H^2 \ominus \Theta H^2$, where $\Theta$ is an inner function. Our results, phrased in terms of the functions that do the interpolation as well as Carleson measures, show that under the assumption that $\Theta(\lambda_n) \to 0$ the interpolation properties in $H^2$ are essentially the same as those in $K_\Theta$.