Reproducing kernels and Riccati equations

Harry Dym

International Journal of Applied Mathematics and Computer Science (2001)

  • Volume: 11, Issue: 1, page 35-53
  • ISSN: 1641-876X

Abstract

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The purpose of this paper is to exhibit a connection between the Hermitian solutions of matrix Riccati equations and a class of finite dimensional reproducing kernel Krein spaces. This connection is then exploited to obtain minimal factorizations of rational matrix valued functions that are J-unitary on the imaginary axis in a natural way.

How to cite

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Dym, Harry. "Reproducing kernels and Riccati equations." International Journal of Applied Mathematics and Computer Science 11.1 (2001): 35-53. <http://eudml.org/doc/207504>.

@article{Dym2001,
abstract = {The purpose of this paper is to exhibit a connection between the Hermitian solutions of matrix Riccati equations and a class of finite dimensional reproducing kernel Krein spaces. This connection is then exploited to obtain minimal factorizations of rational matrix valued functions that are J-unitary on the imaginary axis in a natural way.},
author = {Dym, Harry},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {-inner matrixvalued functions; Riccati equations; Lyapunov equations; reproducing kernel spaces; -unitary matrix valued functions; de Branges spaces; -inner matrix-valued functions; $J$-unitary matrix-valued functions},
language = {eng},
number = {1},
pages = {35-53},
title = {Reproducing kernels and Riccati equations},
url = {http://eudml.org/doc/207504},
volume = {11},
year = {2001},
}

TY - JOUR
AU - Dym, Harry
TI - Reproducing kernels and Riccati equations
JO - International Journal of Applied Mathematics and Computer Science
PY - 2001
VL - 11
IS - 1
SP - 35
EP - 53
AB - The purpose of this paper is to exhibit a connection between the Hermitian solutions of matrix Riccati equations and a class of finite dimensional reproducing kernel Krein spaces. This connection is then exploited to obtain minimal factorizations of rational matrix valued functions that are J-unitary on the imaginary axis in a natural way.
LA - eng
KW - -inner matrixvalued functions; Riccati equations; Lyapunov equations; reproducing kernel spaces; -unitary matrix valued functions; de Branges spaces; -inner matrix-valued functions; $J$-unitary matrix-valued functions
UR - http://eudml.org/doc/207504
ER -

References

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  1. Alpay D. and Dym H. (1986): On applications of reproducingkernel spaces to the Schur algorithm and rational J unitaryfactorization, In: Schur Methods in Operator Theory and Signal Processing (I. Gohberg, Eds.). - Basel: Birkhauser, Vol.OT18, pp.89-159. 
  2. Alpay D. and Dym H. (1993a): On a newclass of structured reproducing kernel spaces. - J. Funct.Anal., Vol.111, No.1, pp.1-28. Zbl0813.46018
  3. Alpay D. and Dym H. (1993b): On a newclass of reproducing kernel spaces and a new generalization of the Iohvidov laws. - Linear Algebra Appl., Vol.178, pp.109-183. Zbl0807.46026
  4. Alpay D. and Dym H. (1996): On a newclass of realization formulas and their application. - Linear Algebra Appl., Vol.241-243, pp.3-84. 
  5. Ball J.A. (1975): Models for noncontractions. - J. Math. Anal. Appl., Vol.52, No.2, pp.240-254. Zbl0317.47004
  6. de Branges L. (1963): Some Hilbert spaces of analytic functions I. - Trans. Amer. Math. Soc., Vol.106, pp.445-468. Zbl0115.33501
  7. Brodskii M.S. (1971): Triangular and Jordan Representations of Linear Operators. - Transl. Math. Monographs., Vol.32, Providence, RI: Amer. Math. Soc. 
  8. Dym H. (1989a): J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation. - CBMS Reg. Conf., Ser. in Math., Vol.71, Providence, RI: Amer. Math. Soc. Zbl0691.46013
  9. Dym H. (1989b): On reproducing kernel spaces, J unitary matrixfunctions, interpolation and displacement rank, In: The Gohberg Anniversary Collection II (H. Dym, S. Goldberg, M.A. Kashoek and P. Lancaster, Eds.). - Oper. Theory Adv. Appl., Vol.OT41, Basel: Birkhauser, pp.173-239. 
  10. Dym H. (1994): Shifts, realizations and interpolation, redux, In: Operator Theory and its Applications (A. Feintuch and I. Gohberg, Eds.). - Oper. Theory Adv. Appl., Vol.OT73, Basel: Birkhauser, pp.182-243. Zbl0896.47010
  11. Dym H. (1998): A basic interpolation problem, In: Holomorphic Spaces, (S. Axler, J.E. Mc Carthy and D. Sarason, Eds.). -Cambridge: Cambridge University Press, pp.381-425. Zbl1128.47305
  12. Dym H. (2001): On Riccati equations and reproducing kernelspaces. - Oper. Theory Adv. Appl., to appear (alsoavailable at t http: www.wisdom.weizmann.ac.il). Zbl0989.46015
  13. Fuhrmann P.A. (1995): On the characterization and parametrization ofminimal spectral factors. - J. Math. Syst. Estim. Contr., Vol.5, No.4, pp.383-441. Zbl0848.30023
  14. Gombani A. and Weiland S. (2000): On the use of J-spectralfactorizations for dissipative dynamical systems. - Proc. 14-th Int. Conf. Math. Theory of Networks and Systems, MTNS, Perpignan, France, published on CD-ROM. 
  15. Karelin I., Lerer L. and Ran A.C.M. (2001): J-symmetric factorizations andalgebraic Riccati equations. - Oper. Theory Adv. Appl., (to appear). Zbl0994.47021
  16. Lerer L. and Ran A.C.M. (1997): J-pseudo-spectral and J-inner-pseudo-outerfactorizations for matrix polynomials. - Int. Eqns.Operat. Theory, Vol.29, pp.23-51. Zbl0896.47015
  17. Rovnyak J. (1968): Characterization of spaces K(M). - (unpublished manuscript). 
  18. Zhou K., Doyle J.C. and Glover K. (1996): Robust and Optimal Control. - New Jersey: Prentice Hall. Zbl0999.49500

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