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

Fricain, Emmanuel; Mashreghi, Javad

Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 2113-2135.

Dans cet article, nous donnons une formule intégrale pour la valeur au bord des dérivées des fonctions de l’espace de de Branges-Rovnyak $\mathscr{H}\left(b\right)$, où $b$ est une fonction dans la boule unité de $H^{\infty }\left({ℂ}_{+}\right)$. En particulier, nous généralisons un résultat d’Ahern-Clark obtenu pour les fonctions de l’espace modèle $K_b$, où $b$ est une fonction intérieure. En utilisant les séries hypergéométriques, nous obtenons une formule non-triviale de combinatoire concernant la somme de coefficients binômiaux. Puis, nous appliquons cette formule pour démontrer que le noyau reproduisant $k_{\omega ,n}^b$, correspondant à l’évaluation de la dérivée $n$-ième des fonctions de $\mathscr{H}\left(b\right)$ au point $\omega$, converge en norme lorsque $\omega$ tend radialement vers un point de l’axe réel.

In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges–Rovnyak spaces $\mathscr{H}\left(b\right)$, where $b$ is in the unit ball of $H^{\infty }\left({ℂ}_{+}\right)$. 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 evaluation of the $n$-th derivative of elements of $\mathscr{H}\left(b\right)$ at the point $\omega$ as it tends radially to a point of the real axis.

Classification : 46E22, 47A15, 33C05, 05A19
Keywords: De Branges-Rovnyak spaces, model subspaces of $H^2$, integral representation, hypergeometric functions
Mot clés : espaces de Branges-Rovnyak, sous-espaces modèle de $H^2$, représentation intégrale, fonctions hypergéométriques

[1] Ahern, P. R.; Clark, D. N. Radial limits and invariant subspaces, Amer. J. Math., Volume 92 (1970), pp. 332-342 | DOI | MR | Zbl

[2] Ahern, P. R.; Clark, D. N. Radial $n\mathrm{th}$ derivatives of Blaschke products, Math. Scand., Volume 28 (1971), pp. 189-201 | MR | Zbl

[3] Anderson, J. M.; Rovnyak, J. On generalized Schwarz-Pick estimates, Mathematika, Volume 53 (2006) no. 1, p. 161-168 (2007) | DOI | MR | Zbl

[4] Andrews, George E.; Askey, Richard; Roy, Ranjan Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999 | MR | Zbl

[5] Baranov, A. D. Bernstein-type inequalities for shift-coinvariant subspaces and their applications to Carleson embeddings, J. Funct. Anal., Volume 223 (2005) no. 1, pp. 116-146 | DOI | MR | Zbl

[6] Bolotnikov, Vladimir; Kheifets, Alexander A higher order analogue of the Carathéodory-Julia theorem, J. Funct. Anal., Volume 237 (2006) no. 1, pp. 350-371 | DOI | MR | Zbl

[7] de Branges, Louis; Rovnyak, James Canonical models in quantum scattering theory, Perturbation Theory and its Applications in Quantum Mechanics (Proc. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Inst., Univ. of Wisconsin, Madison, Wis., 1965), Wiley, New York, 1966, pp. 295-392 | MR | Zbl

[8] de Branges, Louis; Rovnyak, James Square summable power series, Holt, Rinehart and Winston, New York, 1966 | MR

[9] Duren, Peter L. Theory of $H^p$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York, 1970 | MR | Zbl

[10] Dyakonov, Konstantin M. Entire functions of exponential type and model subspaces in $H^p$, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Volume 190 (1991) no. Issled. po Linein. Oper. i Teor. Funktsii. 19, p. 81-100, 186 | MR | Zbl

[11] Dyakonov, Konstantin M. Differentiation in star-invariant subspaces. I. Boundedness and compactness, J. Funct. Anal., Volume 192 (2002) no. 2, pp. 364-386 | DOI | MR | Zbl

[12] Fatou, P. Séries trigonométriques et séries de Taylor, Acta Math., Volume 30 (1906) no. 1, pp. 335-400 | DOI | MR

[13] Fricain, E. Bases of reproducing kernels in de Branges spaces, J. Funct. Anal., Volume 226 (2005) no. 2, pp. 373-405 | DOI | MR | Zbl

[14] Fricain, E.; Mashreghi, J. Boundary behavior of functions of the de Branges-Rovnyak spaces (to appear in Complex Analysis and Operator Theory)

[15] Hartmann, Andreas; Sarason, Donald; Seip, Kristian Surjective Toeplitz operators, Acta Sci. Math. (Szeged), Volume 70 (2004) no. 3-4, pp. 609-621 | MR | Zbl

[16] Helson, Henry Lectures on invariant subspaces, Academic Press, New York, 1964 | MR | Zbl

[17] Jury, Michael T. Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators, Proc. Amer. Math. Soc., Volume 135 (2007) no. 11, p. 3669-3675 (electronic) | DOI | MR | Zbl

[18] Sarason, Donald Sub-Hardy Hilbert spaces in the unit disk, University of Arkansas Lecture Notes in the Mathematical Sciences, 10, John Wiley & Sons Inc., New York, 1994 (A Wiley-Interscience Publication) | MR

[19] Shapiro, Jonathan E. Relative angular derivatives, J. Operator Theory, Volume 46 (2001) no. 2, pp. 265-280 | MR | Zbl

[20] Shapiro, Jonathan E. More relative angular derivatives, J. Operator Theory, Volume 49 (2003) no. 1, pp. 85-97 | MR | Zbl

[21] Slater, Lucy Joan Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966 | MR | Zbl