Complétude des noyaux reproduisants dans les espaces modèles

Fricain, Emmanuel

Annales de l'Institut Fourier, Tome 52 (2002), p. 661-686 / Harvested from Numdam

Résumé

Soit $(\lambda _n)_{n\ge 1}$ une suite de Blaschke du disque unité ${\mathbb {D}}$ et $\Theta $ une fonction intérieure. On suppose que la suite de noyaux reproduisants $\Big (k_\Theta (z,\lambda _n):= {1-\overline{\Theta (\lambda _n)}\Theta (z)\over 1- \overline{\lambda _n}z}\Big )_{n\ge 1}$ est complète dans l’espace modèle $K_\Theta ^p:=H^p\cap \Theta \overline{H^p_0}$, $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 $(\lambda _n)_{n\ge 1}$ mais également sous l’effet de perturbations de la fonction $\Theta $. On retrouve ainsi un certain nombre de résultats classiques sur les systèmes d’exponentielles. Puis, si on suppose de plus que la suite $(k_\Theta (.,\lambda _n))_{n\ge 1}$ est minimale, on montre que, pour une certaine classe de fonctions $\Theta $, la famille biorthogonale associée est aussi complète.

Abstract

Let $(\lambda _n)_{n\ge 1}$ be a Blaschke sequence of the unit disc ${\mathbb {D}}$ and $\Theta $ be an inner function. Assume that the sequence of reproducing kernels $\Big (k_\Theta (z,\lambda _n):= {1-\overline{\Theta (\lambda _n)}\Theta (z)\over 1- \overline{\lambda _n}z}\Big )_{n\ge 1}$ is complete in the model space $K_\Theta ^p:=H^p\cap \Theta \overline{H^p_0}$, $1<p<+\infty $. First of all, we study the stability of this completeness not only under perturbations of frequencies $(\lambda _n)_{n\ge 1}$ but also under perturbations of function $\Theta $. We recover some classical results on exponential systems. Then, if we assume further that the sequence $(k_\Theta (.,\lambda _n))_{n\ge 1}$ is minimal, we show that, for a certain class of functions $\Theta $, the biorthogonal family is also complete.

Détail

Publié le : 2002-01-01

MR 1906486 | Zbl 1032.46040 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1897
Classification:  46E22,  30C40,  30D55,  47A15,  47B32,  47B38
Mots clés: espaces de Hardy, noyaux reproduisants, complétude, systèmes d'exponentielles

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