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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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