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

Emmanuel Fricain[1]; Javad Mashreghi[2]

  • [1] Université Lyon 1 Institut Camille Jordan CNRS UMR 5208 43, boulevard du 11 Novembre 1918 69622 Villeurbanne (France)
  • [2] Université Laval Département de Mathématiques et de Statistique Québec, QC 61VOA6 (Canada)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 6, page 2113-2135
  • ISSN: 0373-0956

Abstract

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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 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 ω , n b of evaluation of the n -th derivative of elements of ℋ ( b ) at the point ω as it tends radially to a point of the real axis.

How to cite

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Fricain, Emmanuel, and Mashreghi, Javad. "Integral representation of the $n$-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel." Annales de l’institut Fourier 58.6 (2008): 2113-2135. <http://eudml.org/doc/10372>.

@article{Fricain2008,
abstract = {In this paper, 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 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.},
affiliation = {Université Lyon 1 Institut Camille Jordan CNRS UMR 5208 43, boulevard du 11 Novembre 1918 69622 Villeurbanne (France); Université Laval Département de Mathématiques et de Statistique Québec, QC 61VOA6 (Canada)},
author = {Fricain, Emmanuel, Mashreghi, Javad},
journal = {Annales de l’institut Fourier},
keywords = {De Branges-Rovnyak spaces; model subspaces of $H^2$; integral representation; hypergeometric functions; de Branges-Rovnyak spaces; model subspaces of },
language = {eng},
number = {6},
pages = {2113-2135},
publisher = {Association des Annales de l’institut Fourier},
title = {Integral representation of the $n$-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel},
url = {http://eudml.org/doc/10372},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Fricain, Emmanuel
AU - Mashreghi, Javad
TI - Integral representation of the $n$-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 6
SP - 2113
EP - 2135
AB - In this paper, 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 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.
LA - eng
KW - De Branges-Rovnyak spaces; model subspaces of $H^2$; integral representation; hypergeometric functions; de Branges-Rovnyak spaces; model subspaces of
UR - http://eudml.org/doc/10372
ER -

References

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