Stability of the bases and frames reproducing kernels in model spaces

Baranov, Anton

Annales de l'Institut Fourier, Tome 55 (2005) no. 7, pp. 2399-2422.

On étudie les bases et les frames des noyaux reproduisants dans les sous-espaces modèles $K_{\Theta }^2=H^2\ominus \Theta H^2$ 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 $k_{{\lambda }_n}\left(z\right)=(1-\overline{\Theta \left({\lambda }_n\right)}\Theta \left(z\right))/(z-{\overline{\lambda }}_n)$ par rapport aux $\langle \langle$ petites $\rangle \rangle$ perturbations des pôles ${\overline{\lambda }}_n$. En utilisant les majorations récentes des derivées dans les espaces $K_{\Theta }^2$, on obtient les estimations des perturbations admissibles, qui généralisent les théorèmes de W.S. Cohn et E. Fricain.

We study the bases and frames of reproducing kernels in the model subspaces $K_{\Theta }^2=H^2\ominus \Theta H^2$ of the Hardy class $H2$ in the upper half-plane. The main problem under consideration is the stability of a basis of reproducing kernels $k_{{\lambda }_n}\left(z\right)=(1-\overline{\Theta \left({\lambda }_n\right)}\Theta \left(z\right))/(z-{\overline{\lambda }}_n)$ 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 $K_{\Theta }^2$ and produce estimates of admissible perturbations generalizing certain results of W.S. Cohn and E. Fricain.

Classification : 46E22, 42C15, 30D55, 47B32
Keywords: Inner function, shift-coinvariant subspace, reproducing kernel, Riesz basis, frame, stability, Inner function, shift-coinvariant subspace, reproducing kernel, Riesz basis, frame, stability
Mot clés : fonction intérieure, espace modèle, noyaux reproduisant, base de Riesz, frame, stabilité

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