Abstract
In this paper we consider the problem of building the spectral function of a canonical differential equation when the potential is given. We restrict ourselves to the case where the spectral function is rational. An algorithm is proposed which allows the construction of the spectral function from the values of the potential and of a number of its derivatives at the origin. The approach is based on the solution of the partial realization problem for systems.
Dedicated to the memory of I.M. Glazman
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Alpay and I. Gohberg. Unitary rational matrix functions, volume 33 of Operator Theory: Advances and Applications, pages 175–222. Birkhäuser Verlag, Basel, 1988.
D. Alpay and I. Gohberg. Inverse spectral problem for differential operators with rational scattering matrix functions. Journal of Differential Equations, 118:1–19, 1995.
H. Bart, I. Gohberg, and M. Kaashoek. Minimal factorization of matrix and operator functions, volume 1 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1979.
H. Bart, I. Gohberg, and M. Kaashoek. Convolution equations and linear systems. Integral Equations and Operator Theory, 5:283–340, 1982.
A. Bruckstein, B. Levy, and T. Kailath. Differential methods in inverse scattering. SIAM journal of applied mathematics, 45:312–335, 1985.
H. Dym and A. Iacob. Positive definite extensions, canonical equations and inverse problems, volume 12 of Operator Theory: Advances and Applications, pages 141–240. Birkhäuser Verlag, Basel, 1984.
I. Gohberg, M. Kaashoek, and L. Lerer. On minimality in the partial realization problem. Systems and Control Letters, 9:97–104, 1987.
M.G. KreÄn. Continuous analogues of propositions for polynomials orthogonal on the unit circle. Dokl. Akad. Nauk. SSSR, 105:637–640, 1955.
M.G. KreÄn and F.E. Melik-Adamyan. On the theory of 5-matrices of canonical equations with summable potentials. Dokl. Akad. Nauk. SSSR, 16:150–159, 1968.
F.E. Melik-Adamyan. Canonical differential operators in Hilbert space. Izvestya Akademii Nauk. Armyanskoi SSR Matematica, 12:10–31, 1977.
F.E. Melik-Adamyan. On a class of canonical differential operators. Izvestya Akademii Nauk. Armyanskoi SSR Matematica, 24:570–592, 1989. English translation in: Soviet Journal of Contemporary Mathematics, vol. 24, pages 48-69 (1989).
M.W. Wonham. Linear Multivariable Control: Geometric Approach. Springer-Verlag, New-York, 1979.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Basel AG
About this paper
Cite this paper
Alpay, D., Gohberg, I. (1997). Potentials Associated to Rational Weights. In: Gohberg, I., Lyubich, Y. (eds) New Results in Operator Theory and Its Applications. Operator Theory: Advances and Applications, vol 98. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8910-0_3
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8910-0_3
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9824-9
Online ISBN: 978-3-0348-8910-0
eBook Packages: Springer Book Archive