In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges–Rovnyak spaces , where is in the unit ball of . In particular, we generalize a result of Ahern–Clark obtained for functions of the model spaces , where 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 of evaluation of the -th derivative of elements of at the point 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 , où est une fonction dans la boule unité de . En particulier, nous généralisons un résultat d’Ahern-Clark obtenu pour les fonctions de l’espace modèle , où 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 , correspondant à l’évaluation de la dérivée -ième des fonctions de au point , converge en norme lorsque tend radialement vers un point de l’axe réel.
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
@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}, pages = {2113--2135}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {58}, number = {6}, year = {2008}, doi = {10.5802/aif.2408}, zbl = {1159.46016}, mrnumber = {2473631}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2408/} }
TY - JOUR AU - Fricain, Emmanuel AU - Mashreghi, Javad TI - Integral representation of the $n$-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel JO - Annales de l'Institut Fourier PY - 2008 SP - 2113 EP - 2135 VL - 58 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2408/ DO - 10.5802/aif.2408 LA - en ID - AIF_2008__58_6_2113_0 ER -
%0 Journal Article %A Fricain, Emmanuel %A Mashreghi, Javad %T Integral representation of the $n$-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel %J Annales de l'Institut Fourier %D 2008 %P 2113-2135 %V 58 %N 6 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2408/ %R 10.5802/aif.2408 %G en %F 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, Volume 58 (2008) no. 6, pp. 2113-2135. doi : 10.5802/aif.2408. http://archive.numdam.org/articles/10.5802/aif.2408/
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