Abstract
An overview is given of some classical and recent results concerning zeros of orthogonal matrix polynomials on the unit circle. The basic questions are: How these zeros are located in the complex plane? Conversely, what conditions on the location of the zeros of a given matrix polynomial ensure that the polynomial is orthogonal?
This Material is based upon research supported in part by the National Science Foundation under grant number DMS 88-02836
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
N. I. Akhiezer, The Classical Moment Problem, Hafner Publishing Company, New York, 1965.
D. Alpay, I. Gohberg, On orthogonal matrix polynomials, Operator Theory: Advances and Applications 34 (1988), 25 – 46.
J.A. Ball, I. Gohberg, and L. Rodman, Interpolation Problems for Rational Matrix Functions, monograph to be published by Birkhäuser.
H. Bart, I. Gohberg and M.A. Kaashoek, Minimal Factorizations of Matrix and Operator Functions, Birkhäuser, Basel, 1979.
A. Ben-Artzi and I. Gohberg, Extension of a theorem of M. G. Krein on orthogonal polynomials for the nonstationary case, Operator Theory: Advances and Applications 34 (1988), 65 – 78.
P. Delsarte and Y. Genin, Spectral Properties of Finite Toeplitz Matrices, Mathematical Theory of Networks and Systems (Paul A. Fuhrmann, ed.), Lecture Notes in Control and Information Sciences, No. 58, Springer-Verlag, New York (1984), 194 – 213.
Ph. Delsarte, Y. Genin and Y. Kamp, Orthogonal polynomial matrices on the unit circle, IEEE Trans. Circuits & Systems 25 (1978), 145 – 160.
Ph. Delsarte, Y. Genin and Y. Kamp, Schur parametrization of positive definite block-Toeplitz systems, SIAM J. Applied Math. 36 (1979), 34 – 46.
H. Dym, Hermitian Block Toeplitz matrices, orthogonal polynomials, reproducing kernel Pontryagin spaces, interpolation and extension, Operator Theory: Advances and Applications 34 (1988), 79 – 135.
R. L. Ellis, I. Gohberg and D. C. Lay,. On two theorems of M. G. Krein concerning polynomials orthogonal on the unit circle, Integral Equations and Operator Theory 11 (1988), 87 – 104.
B. Fritzsche and B. Kirstein, An extension problem for non-negative Hermitian block Toeplitz matrices HI, Math Nachr. 135 (1988), 319 – 341.
P. A. Fuhrmann, Orthogonal matrix polynomials and system theory, Preprint, 1986.
J. S. Geronimo, Matrix orthogonal polynomials on the unit circle, J. Math. Phys. 22 (7) (1981), 1359 – 1365.
L. Ya. Geronimus, Orthogonal Polynomials, New York, Consultants Bureau, 1961.
I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, 1982.
I. Gohberg, P. Lancaster and L. Rodman, Invariant Subspaces of Matrices with Applications, J. Wiley, New York, 1986.
I. Gohberg and L. Lerer, Matrix generalizations of M. G. Krein theorems on orthogonal polynomials, Operator Theory: Advances and Applications 34(1988).
T. Kailath, A view of three decades of linear filtering theory, IEEE trans. Information Theory 20 (1974), 145 – 181.
T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980.
T. Kailath, A. Vieira and M. Morf, Inverses of Toeplitz operators, innovations, and orthogonal polynomials. SIAM Review 20 (1978), 106 – 119.
M. G. Krein, On the Distribution of the Roots of Polynomials which are Orthogonal on the Unit Circle with Respect to an Alternating Weight, Teor. Funkciĭ Funkcional Anal, i Priložen. Resp. Sb. Nr. 2(1966), 131–137 [Russian].
H. J. Landau, Maximum entropy and the moment problem, Bull. Amer. Math. Soc. 16 (1987), 47 – 77.
M. Rosenberg, The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure, Duke Math. J. 31 (1964), 291 – 298.
G. Szegö, Orthogonal Polynomial, Colloquium Publications, No. 23, Amer. Math. Soc., Providence, RI 2nd ed. 1958, 3rd ed. 1967.
J. V. Uspensky, Theory of Equations, McGraw-Hill, New York, 1978.
D. C. Youla and N. Kazanjian,. Bauer-type factorization of positive matrices and the theory of matrix polynomials orthogonal on the unit circle, IEEE Trans, on Circuits and Systems, CAS-25(1978), 57–69.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Kluwer Academic Publishers
About this chapter
Cite this chapter
Rodmank, L. (1990). Orthogonal Matrix Polynomials. In: Nevai, P. (eds) Orthogonal Polynomials. NATO ASI Series, vol 294. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0501-6_16
Download citation
DOI: https://doi.org/10.1007/978-94-009-0501-6_16
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6711-9
Online ISBN: 978-94-009-0501-6
eBook Packages: Springer Book Archive