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Bases of reproducing kernels in de-Branges spaces

Bases of reproducing kernels in de Branges spaces
Authors: Fricain, E.;

Bases of reproducing kernels in de-Branges spaces

Abstract

This paper deals with geometric properties of sequences of reproducing kernels related to de-Branges spaces. If $b$ is a nonconstant function in the unit ball of $H^\infty$ , and $T_b$ is the Toeplitz operator, with symbol $b$, then the de-Branges space, $H(b)$, associated to $b$, is defined by $H(b)=(Id−T_b^* T_b)^{1/2}H^2$ , where $H^2$ is the Hardy space of the unit disk. It is equiped with the inner product such that $(Id −T_b^* T_b)^{1/2}$ is a partial isometry from $H^2$ onto $H(b)$. First, following a work of Ahern-Clark, we study the problem of orthogonal basis of reproducing kernels in $H(b)$. Then we give a criterion for sequences of reproducing kernels which form an unconditionnal basis in their closed linear span. As far as concerns the problem of complete unconditionnal basis in $H(b)$, we show that there is a dichotomy between the case where $b$ is an extreme point of the unit ball of $H^\infty$ and the opposite case.

Country
France
Keywords

Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces), de Branges spaces, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, reproducing kernels., Riesz bases, [MATH] Mathematics [math], de-Branges spaces, Reproducing kernels, reproducing kernels, Banach algebras of differentiable or analytic functions, \(H^p\)-spaces, Linear operators on function spaces (general), Spaces of bounded analytic functions of one complex variable, Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces), de Branges--Rovnyak spaces, Analysis

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
11
Top 10%
Top 10%
Average
Green
hybrid