Geometry of reproducing kernels in model spaces near the boundary

TL;DR

This paper investigates geometric properties of reproducing kernels in model spaces near the boundary, focusing on overcompleteness and minimal systems related to Ahern--Clark points, revealing new conditions for quasi-analyticity.

Contribution

It establishes the existence of non-Riesz minimal sequences near certain boundary points and links overcompleteness to zero localization, advancing understanding of boundary behavior in model spaces.

Findings

Non-Riesz minimal sequences exist near non-analytic Ahern--Clark points.

Overcompleteness occurs only near infinite order Ahern--Clark points.

Conditions for quasi-analyticity of model space restrictions are provided.

Abstract

We study two geometric properties of reproducing kernels in model spaces $K\_\theta$where $\theta$ is an inner function in the disc: overcompleteness and existence of uniformly minimalsystems of reproducing kernels which do not contain Riesz basic sequences. Both of these properties are related to the notion of the Ahern--Clark point. It is shown that "uniformly minimal non-Riesz"$ $ sequences of reproducing kernelsexist near each Ahern--Clark point which is not an analyticity point for $\theta$, whileovercompleteness may occur only near the Ahern--Clark points of infinite orderand is equivalent to a "zero localization property". In this context the notion ofquasi-analyticity appears naturally, and as a by-product of our results we give conditions in thespirit of Ahern--Clark for the restriction of a model space to a radius to be a class ofquasi-analyticity.