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

TL;DR

This paper provides an integral representation for derivatives in de Branges-Rovnyak spaces, generalizes existing results, and proves the norm convergence of their reproducing kernels using hypergeometric series and combinatorial formulas.

Contribution

It introduces a new integral formula for boundary derivatives in de Branges-Rovnyak spaces and demonstrates the norm convergence of associated reproducing kernels.

Findings

Integral representation for boundary derivatives in de Branges-Rovnyak spaces.

Generalization of Ahern--Clark's result to non-inner functions.

Proof of norm convergence of reproducing kernels as points approach the boundary.

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.