Integral representation of the n-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel  [ Représentation intégrale pour la dérivée n-ième des fonctions de l’espace de de Branges-Rovnyak et la convergence en norme de son noyau reproduisant ]
Annales de l'Institut Fourier, Tome 58 (2008) no. 6, p. 2113-2135
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.
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.
DOI : https://doi.org/10.5802/aif.2408
Classification:  46E22,  47A15,  33C05,  05A19
Mots clés: espaces de Branges-Rovnyak, sous-espaces modèle de H 2 , représentation intégrale, fonctions hypergéométriques
@article{AIF_2008__58_6_2113_0,
     author = {Fricain, Emmanuel and Mashreghi, Javad},
     title = {Integral representation of the $n$-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {6},
     year = {2008},
     pages = {2113-2135},
     doi = {10.5802/aif.2408},
     mrnumber = {2473631},
     zbl = {1159.46016},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_6_2113_0}
}
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, Tome 58 (2008) no. 6, pp. 2113-2135. doi : 10.5802/aif.2408. https://www.numdam.org/item/AIF_2008__58_6_2113_0/

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