Complétude des noyaux reproduisants dans les espaces modèles

Fricain, Emmanuel

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

Soit ${\left({\lambda }_n\right)}_{n\ge 1}$ une suite de Blaschke du disque unité $𝔻$ et $\Theta$ une fonction intérieure. On suppose que la suite de noyaux reproduisants ${\left(k_{\Theta }(z,{\lambda }_n):=\frac{1-\overline{\Theta \left({\lambda }_n\right)}\Theta \left(z\right)}{1-\overline{{\lambda }_n}z}\right)}_{n\ge 1}$ est complète dans l’espace modèle $K_{\Theta }^p:=H^p\cap \Theta \overline{H_0^p}$, $1<p<+\infty$. On étudie, dans un premier temps, la stabilité de cette propriété de complétude, à la fois sous l’effet de perturbations des fréquences ${\left({\lambda }_n\right)}_{n\ge 1}$ 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 ${\left(k_{\Theta }(.,{\lambda }_n)\right)}_{n\ge 1}$ est minimale, on montre que, pour une certaine classe de fonctions $\Theta$, la famille biorthogonale associée est aussi complète.

Let ${\left({\lambda }_n\right)}_{n\ge 1}$ be a Blaschke sequence of the unit disc $𝔻$ and $\Theta$ be an inner function. Assume that the sequence of reproducing kernels ${\left(k_{\Theta }(z,{\lambda }_n):=\frac{1-\overline{\Theta \left({\lambda }_n\right)}\Theta \left(z\right)}{1-\overline{{\lambda }_n}z}\right)}_{n\ge 1}$ is complete in the model space $K_{\Theta }^p:=H^p\cap \Theta \overline{H_0^p}$, $1<p<+\infty$. First of all, we study the stability of this completeness not only under perturbations of frequencies ${\left({\lambda }_n\right)}_{n\ge 1}$ but also under perturbations of function $\Theta$. We recover some classical results on exponential systems. Then, if we assume further that the sequence ${\left(k_{\Theta }(.,{\lambda }_n)\right)}_{n\ge 1}$ is minimal, we show that, for a certain class of functions $\Theta$, the biorthogonal family is also complete.

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

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