Stability of the bases and frames reproducing kernels in model spaces

Baranov, Anton

doi:10.5802/aif.2165

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 ¹

¹ Université Bordeaux 1, Laboratoire d'Analyse et Géométrie, 351 cours de la Libération, 33405 Talence (France), Institutionen för Matematik, Kgl Tekniska Högskolan, 100 44 Stockholm (Suède)

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

Résumé
Abstract

On étudie les bases et les frames des noyaux reproduisants dans les sous-espaces modèles $K_{Θ}^{2} = H^{2} ⊖ Θ H^{2}$ 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 $k_{λ_{n}} (z) = (1 - \bar{Θ (λ_{n})} Θ (z)) / (z - {\bar{λ}}_{n})$ par rapport aux $〈 〈$ petites $〉 〉$ perturbations des pôles ${\bar{λ}}_{n}$ . En utilisant les majorations récentes des derivées dans les espaces $K_{Θ}^{2}$ , 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 $K_{Θ}^{2} = H^{2} ⊖ Θ H^{2}$ of the Hardy class $H 2$ in the upper half-plane. The main problem under consideration is the stability of a basis of reproducing kernels $k_{λ_{n}} (z) = (1 - \bar{Θ (λ_{n})} Θ (z)) / (z - {\bar{λ}}_{n})$ under “small” perturbations of the points $λ_{n}$ . We propose an approach to this problem based on the recently obtained estimates of derivatives in the spaces $K_{Θ}^{2}$ and produce estimates of admissible perturbations generalizing certain results of W.S. Cohn and E. Fricain.

MR Zbl

DOI : 10.5802/aif.2165

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

Affiliations des auteurs :

Baranov, Anton ¹

¹ Université Bordeaux 1, Laboratoire d'Analyse et Géométrie, 351 cours de la Libération, 33405 Talence (France), Institutionen för Matematik, Kgl Tekniska Högskolan, 100 44 Stockholm (Suède)

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     title = {Stability of the bases and frames reproducing kernels in model spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {2399--2422},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {7},
     year = {2005},
     doi = {10.5802/aif.2165},
     mrnumber = {2207388},
     zbl = {1101.30036},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.2165/}
}

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Baranov, Anton. Stability of the bases and frames reproducing kernels in model spaces. Annales de l'Institut Fourier, Tome 55 (2005) no. 7, pp. 2399-2422. doi : 10.5802/aif.2165. https://www.numdam.org/articles/10.5802/aif.2165/

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