Thin Sequences and Their Role in Model Spaces and Douglas Algebras

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 $K_{\Theta }$.

v1: 16 pages. v2: 18 pages, corrections and modifications based on referee comments. To appear in J. Fourier Anal. Appl