Integral representation]{Integral representation of the $n$-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel

Emmanuel Fricain (1) , Javad Mashreghi (2)

1 ICJ - Institut Camille Jordan
2 Université LAVAL, Département de Mathématiques et de Statistiques

In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges--Rovnyak spaces $\HH(b)$, where $b$ is in the unit ball of $H^\infty(\CC_+)$. 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 $k_{\omega,n}^b$ of the evaluation of $n$-th derivative of elements of $\HH(b)$ at the point $\omega$ as it tends radially to a point of the real axis.

Domaines: Variables complexes [math.CV]; Combinatoire [math.CO]; Analyse fonctionnelle [math.FA]

Emmanuel Fricain, Javad Mashreghi. Integral representation]{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, 2008, 58 (6), pp.2113-2135. ⟨10.5802/aif.2408⟩. ⟨hal-00241328⟩