Stability of the bases and frames reproducing kernels in model spaces

[Stabilité de bases et frames des noyaux reproduisants dans les espaces modèles]

Baranov, Anton

Annales de l'Institut Fourier, Tome 55 (2005), p. 2399-2422 / Harvested from Numdam

Résumé

On étudie les bases et les frames des noyaux reproduisants dans les sous-espaces modèles $K^2_{\Theta }=H^2\ominus \Theta H^2$ de l’espace de Hardy $H^2$ dans le demi-plan supérieur. On considère le problème de la stabilité d’une base des noyaux reproduisants $k_{\lambda _n}(z)= (1-\overline{\Theta (\lambda _n)}\Theta (z))/(z-\overline{\lambda }_n)$ par rapport aux $\scriptstyle \langle \!\langle $ petites $\scriptstyle \rangle \!\rangle $ perturbations des pôles $\overline{\lambda }_n$. En utilisant les majorations récentes des derivées dans les espaces ${K^2_{\Theta }}$, on obtient les estimations des perturbations admissibles, qui généralisent les théorèmes de W.S. Cohn et E. Fricain.

Abstract

We study the bases and frames of reproducing kernels in the model subspaces $K^2_{\Theta }=H^2\ominus \Theta H^2$ of the Hardy class $H2$ in the upper half-plane. The main problem under consideration is the stability of a basis of reproducing kernels $k_{\lambda _n}(z)= (1-\overline{\Theta (\lambda _n)}\Theta (z))/(z-\overline{\lambda }_n)$ under “small” perturbations of the points $\lambda _n$. We propose an approach to this problem based on the recently obtained estimates of derivatives in the spaces ${K^2_{\Theta }}$ and produce estimates of admissible perturbations generalizing certain results of W.S. Cohn and E. Fricain.

Détail

Publié le : 2005-01-01

MR 2207388 | Zbl 1101.30036

DOI : https://doi.org/10.5802/aif.2165
Classification:  46E22,  42C15,  30D55,  47B32
Mots clés: fonction intérieure, espace modèle, noyaux reproduisant, base de Riesz, frame, stabilité

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