Integral representation of the n-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel
Annales de l'Institut Fourier, Volume 58 (2008) no. 6, pp. 2113-2135.

In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges–Rovnyak spaces (b), where b is in the unit ball of H ( + ). In particular, we generalize a result of Ahern–Clark obtained for functions of the model spaces K b , where b is an inner function. Using hypergeometric series, we obtain a nontrivial formula of combinatorics for sums of binomial coefficients. Then we apply this formula to show the norm convergence of reproducing kernel k ω,n b of evaluation of the n-th derivative of elements of (b) at the point ω as it tends radially to a point of the real axis.

Dans cet article, nous donnons une formule intégrale pour la valeur au bord des dérivées des fonctions de l’espace de de Branges-Rovnyak (b), où b est une fonction dans la boule unité de H ( + ). En particulier, nous généralisons un résultat d’Ahern-Clark obtenu pour les fonctions de l’espace modèle K b , où b est une fonction intérieure. En utilisant les séries hypergéométriques, nous obtenons une formule non-triviale de combinatoire concernant la somme de coefficients binômiaux. Puis, nous appliquons cette formule pour démontrer que le noyau reproduisant k ω,n b , correspondant à l’évaluation de la dérivée n-ième des fonctions de (b) au point ω, converge en norme lorsque ω tend radialement vers un point de l’axe réel.

DOI: 10.5802/aif.2408
Classification: 46E22, 47A15, 33C05, 05A19
Keywords: De Branges-Rovnyak spaces, model subspaces of $H^2$, integral representation, hypergeometric functions
Mot clés : espaces de Branges-Rovnyak, sous-espaces modèle de $H^2$, représentation intégrale, fonctions hypergéométriques

Fricain, Emmanuel 1; Mashreghi, Javad 2

1 Université Lyon 1 Institut Camille Jordan CNRS UMR 5208 43, boulevard du 11 Novembre 1918 69622 Villeurbanne (France)
2 Université Laval Département de Mathématiques et de Statistique Québec, QC 61VOA6 (Canada)
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Fricain, Emmanuel; Mashreghi, Javad. Integral representation of the $n$-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel. Annales de l'Institut Fourier, Volume 58 (2008) no. 6, pp. 2113-2135. doi : 10.5802/aif.2408. https://aif.centre-mersenne.org/articles/10.5802/aif.2408/

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