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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.
We use reproducing kernel methods to study various rigidity problems. The methods and setting allow us to also consider the non-positive case.
For any real $beta$ let $H^2_beta$ be the Hardy-Sobolev space on the unit disk $D$. $H^2_beta$ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $beta>1/2$. In this paper, we study composition operators $C_varphi$ on $H
The Gaussian kernel plays a central role in machine learning, uncertainty quantification and scattered data approximation, but has received relatively little attention from a numerical analysis standpoint. The basic problem of finding an algorithm fo
We introduce a vector differential operator $mathbf{P}$ and a vector boundary operator $mathbf{B}$ to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This reproducing ker
The subject of this paper is Beurlings celebrated extension of the Riemann mapping theorem cite{Beu53}. Our point of departure is the observation that the only known proof of the Beurling-Riemann mapping theorem contains a number of gaps which seem i