INTEGRAL REPRESENTATION OF THE n TH DERIVATIVE IN DE BRANGES ROVNYAK SPACES AND THE NORM CONVERGENCE

INTEGRAL REPRESENTATION OF THE n-TH DERIVATIVE IN DE BRANGES-ROVNYAK SPACES AND THE NORM CONVERGENCE OF ITS REPRODUCING KERNEL EMMANUEL FRICAIN, JAVAD MASHREGHI Abstract. In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges–Rovnyak spaces H(b), where b is in the unit ball of H∞(C+). In particular, we generalize a result of Ahern–Clark obtained for functions of the model spaces Kb, 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 kb?,n of the evaluation of n-th derivative of elements of H(b) at the point ? as it tends radially to a point of the real axis. 1. Introduction Let C+ denote the upper half plane in the complex plane and let H2(C+) denote the usual Hardy space consisting of analytic functions f on C+ which satisfy ?f?2 := sup y>0 (∫ R |f(x+ iy)|2 dx )1/2 < +∞. P. Fatou [12] proved that, for any function f in H2(C+) and for almost all x0 in R, f?(x0) := lim t?0+ f(x0 + it) exists.

• interesting relations


• hypergeometric functions


• space contractions


• blaschke product


• hilbert space


• radial limits


• schwarz-pick matrix


• branges-rovnyak spaces