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Integral representation of the $$n$$-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel. (English) Zbl 1159.46016
Summary: We give an integral representation for the boundary values of derivatives of functions of the de Branges-Rovnyak spaces $$\mathcal H(b)$$, where $$b$$ is in the unit ball of $$H^\infty(\mathbb C_+)$$. 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 the reproducing kernel $$k_{\omega,n}^b$$ of evaluation of the $$n$$-th derivative of elements of $${\mathcal H}(b)$$ at the point $$\omega$$ as it tends radially to a point of the real axis.

##### MSC:
 46E20 Hilbert spaces of continuous, differentiable or analytic functions 46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) 30D55 $$H^p$$-classes (MSC2000) 47A15 Invariant subspaces of linear operators 33C05 Classical hypergeometric functions, $${}_2F_1$$ 05A19 Combinatorial identities, bijective combinatorics
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