Complétude des noyaux reproduisants dans les espaces modèles  [ Completeness of reproducing kernels in the model spaces ]
Annales de l'Institut Fourier, Volume 52 (2002) no. 2, p. 661-686

Let (λ n ) n1 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 ) : = 1-Θ(λ n ) ¯Θ(z) 1-λ n ¯z n1 is complete in the model space K Θ p :=H p ΘH 0 p ¯, 1<p<+. First of all, we study the stability of this completeness not only under perturbations of frequencies (λ n ) n1 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 )) n1 is minimal, we show that, for a certain class of functions Θ, the biorthogonal family is also complete.

Soit (λ n ) n1 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 ) : = 1-Θ(λ n ) ¯Θ(z) 1-λ n ¯z n1 est complète dans l’espace modèle K Θ p :=H p ΘH 0 p ¯, 1<p<+. 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 ) n1 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 )) n1 est minimale, on montre que, pour une certaine classe de fonctions Θ, la famille biorthogonale associée est aussi complète.

DOI : https://doi.org/10.5802/aif.1897
Classification:  46E22,  30C40,  30D55,  47A15,  47B32,  47B38
Keywords: Hardy spaces, reproducing kernels, completeness, exponential systems
@article{AIF_2002__52_2_661_0,
     author = {Fricain, Emmanuel},
     title = {Compl\'etude des noyaux reproduisants dans les espaces mod\`eles},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {2},
     year = {2002},
     pages = {661-686},
     doi = {10.5802/aif.1897},
     zbl = {1032.46040},
     mrnumber = {1906486},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2002__52_2_661_0}
}
Fricain, Emmanuel. Complétude des noyaux reproduisants dans les espaces modèles. Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 661-686. doi : 10.5802/aif.1897. http://www.numdam.org/item/AIF_2002__52_2_661_0/

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