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

Abstract

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.