Complétude des noyaux reproduisants dans les espaces modèles

Fricain, Emmanuel

doi:10.5802/aif.1897

Fricain, Emmanuel ¹

¹ Université Claude Bernard Lyon I, Institut Girard Desargues, 43 bd du 11 novembre 1918, 69622 Villeurbanne Cedex (France)

Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 661-686.

Résumé
Abstract

Soit ${(λ_{n})}_{n \geq 1}$ une suite de Blaschke du disque unité $𝔻$ et $Θ$ une fonction intérieure. On suppose que la suite de noyaux reproduisants ${(k_{Θ} (z, λ_{n}) : = \frac{1 - \bar{Θ (λ_{n})} Θ (z)}{1 - \bar{λ_{n}} z})}_{n \geq 1}$ est complète dans l’espace modèle $K_{Θ}^{p} : = H^{p} \cap Θ \bar{H_{0}^{p}}$ , $1 < p < + \infty$ . On étudie, dans un premier temps, la stabilité de cette propriété de complétude, à la fois sous l’effet de perturbations des fréquences ${(λ_{n})}_{n \geq 1}$ mais également sous l’effet de perturbations de la fonction $Θ$ . On retrouve ainsi un certain nombre de résultats classiques sur les systèmes d’exponentielles. Puis, si on suppose de plus que la suite ${(k_{Θ} (., λ_{n}))}_{n \geq 1}$ est minimale, on montre que, pour une certaine classe de fonctions $Θ$ , la famille biorthogonale associée est aussi complète.

Let ${(λ_{n})}_{n \geq 1}$ be a Blaschke sequence of the unit disc $𝔻$ and $Θ$ be an inner function. Assume that the sequence of reproducing kernels ${(k_{Θ} (z, λ_{n}) : = \frac{1 - \bar{Θ (λ_{n})} Θ (z)}{1 - \bar{λ_{n}} z})}_{n \geq 1}$ is complete in the model space $K_{Θ}^{p} : = H^{p} \cap Θ \bar{H_{0}^{p}}$ , $1 < p < + \infty$ . First of all, we study the stability of this completeness not only under perturbations of frequencies ${(λ_{n})}_{n \geq 1}$ but also under perturbations of function $Θ$ . We recover some classical results on exponential systems. Then, if we assume further that the sequence ${(k_{Θ} (., λ_{n}))}_{n \geq 1}$ is minimal, we show that, for a certain class of functions $Θ$ , the biorthogonal family is also complete.

MR Zbl

DOI : 10.5802/aif.1897

Classification : 46E22, 30C40, 30D55, 47A15, 47B32, 47B38
Mot clés : espaces de Hardy, noyaux reproduisants, complétude, systèmes d'exponentielles
Keywords: Hardy spaces, reproducing kernels, completeness, exponential systems

Affiliations des auteurs :

Fricain, Emmanuel ¹

¹ Université Claude Bernard Lyon I, Institut Girard Desargues, 43 bd du 11 novembre 1918, 69622 Villeurbanne Cedex (France)

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Fricain, Emmanuel. Complétude des noyaux reproduisants dans les espaces modèles. Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 661-686. doi : 10.5802/aif.1897. https://aif.centre-mersenne.org/articles/10.5802/aif.1897/

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