Stability of the bases and frames reproducing kernels in model spaces

Baranov, Anton

doi:10.5802/aif.2165

Baranov, Anton

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

Abstract
Résumé

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.

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.

MR 2207388 | Zbl 1101.30036

DOI : https://doi.org/10.5802/aif.2165
Classification: 46E22, 42C15, 30D55, 47B32
Keywords: Inner function, shift-coinvariant subspace, reproducing kernel, Riesz basis, frame, stability
Keywords: Inner function, shift-coinvariant subspace, reproducing kernel, Riesz basis, frame, stability

@article{AIF_2005__55_7_2399_0,
     author = {Baranov, Anton},
     title = {Stability of the bases and frames reproducing kernels in model spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {7},
     year = {2005},
     pages = {2399-2422},
     doi = {10.5802/aif.2165},
     zbl = {1101.30036},
     mrnumber = {2207388},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2005__55_7_2399_0}
}

Baranov, Anton. Stability of the bases and frames reproducing kernels in model spaces. Annales de l'Institut Fourier, Volume 55 (2005) no. 7, pp. 2399-2422. doi : 10.5802/aif.2165. http://www.numdam.org/item/AIF_2005__55_7_2399_0/

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