In this paper we obtain explicit formulas for the coefficients of a second order difference block operator if its spectral or its scattering functions are rational matrix functions analytic and invertible on the unit circle. The solutions are given in terms of realizations of the spectral or scattering function.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
D. Alpay, J. Ball, I. Gohberg, and L. Rodman.Realization and factorization of rational matrix functions with summetries, volume 47 ofOperator Theory: Advances and Applications, pages 1–60. Birkhäuser Verlag, Basel, 1990.
D. Alpay, J. Ball, I. Gohberg, and L. Rodman. State space theory of automorphisms of rational matrix functions.Integral Equations and Operator Theory, 15:349–377, 1992.
D. Alpay, J. Ball, I. Gohberg, and L. Rodman.J-unitary preserving automorphisms of rational matrix functions: State space theory, interpolation and factorization.Linear Algebra and its Applications, 197–198:531–566, 1994.
D. Alpay and H. Dym.On applications of reproducing kernel spaces to the Schur algorithm and rational J-unitary factorization, volume 18 ofOperator Theory: Advances and Applications, pages 89–159. Birkhäuser Verlag, Basel, 1986.
D. Alpay and I. Gohberg.Unitary rational matrix functions, volume 33 ofOperator 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. To appear in Journal of Differential Equations, 1993.
J. Ball, I. Gohberg, and L. Rodman.Interpolation of rational matrix functions. Birkhäuser Verlag, Basel, 1990.
J. Ball and A. Ran.Left versus right canonical Wiener-Hopf factorization, volume 21 ofOperator Theory: Advances and Applications, pages 9–38. Birkhäuser-Verlag, Basel, 1986.
H. Bart, I. Gohberg, and M. Kaashoek.Minimal factorization of matrix and operator functions, volume 1 ofOperator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1979.
Yu Berezanskii.Expansions in eigenfunctions of selfadjoint operators, volume 17 ofTranslations of mathematical monographs. American Mathematical Society, 1968.
P. Delsarte and Y. Genin. On a generalization of the Szegö-Levinson recurrence and its application in lossless inverse scattering.IEEE Transactions on Information Theory, 38:104–111, 1992.
P. Delsarte, Y. Genin, and Y. Kamp. Orthogonal polynomial matrices on the unit circle.IEEE Transactions on Circuits and Systems, pages 149–160, 1978.
P. Delsarte, Y. Genin, and Y. Kamp. Schur parametrization of positive definite block-Toeplitz systems.SIAM Journal in Applied Mathematics, 36:34–46, 1979.
H. Dym.Hermitian block Toeplitz matrices, orthogonal polynomials, reproducing kernel Pontryagin spaces, interpolation and extension, volume 34 ofOperator Theory: Advances and Applications, pages 79–135. Birkhäuser Verlag, Basel, 1988.
H. Dym.J-contractive matrix functions, reproducing kernel spaces and interpolation, volume 71 ofCBMS Lecture Notes. Amer. Math. Soc., Rhodes Island, 1989.
H. Dym. On Hermitian block Hankel matrices, matrix polynomials, the Hamburger moment problem, interpolation and maximum entropy.Integral Equations and Operator Theory, 12:757–812, 1989.
H. Dym and A. Iacob.Applications of factorization and Toeplitz operators to inverse problem, volume 4 ofOperator Theory: Advances and Applications, pages 233–260. Birkhäuser Verlag, Basel, 1982.
J. S. Geronimo. Scattering theory and matrix orthogonal polynomials on the real line.Circuits, systems and signal processing, 1:471–495, 1982.
I. Gohberg, M. Kaashoek, and F. van Schagen. Szegö-Kac-Achiezer formulas in terms of realizations of the symbol.Journal of functional analysis, 74:24–51, 1987.
I. Gohberg, P. Lancaster, and L. Rodman.Invariant subspaces of matrices and applications. Wiley, New-York, 1986.
M.W. Wonham.Linear Multivariable Control: Geometric Approach. Springer-Verlag, New-York, 1979.
About this article
Cite this article
Alpay, D., Gohberg, I. Inverse spectral problems for difference operators with rational scattering matrix function. Integr equ oper theory 20, 125–170 (1994). https://doi.org/10.1007/BF01679669
AMS Subject Classification Numbers