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Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory

Published online by Cambridge University Press:  20 November 2018

L. Miranian
L. Miranian*
Affiliation:
Department of Mathematics, University of California, Berkeley CA, 94720 e-mail: luiza@math.berkeley.edu
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Abstract

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In the work presented below the classical subject of orthogonal polynomials on the unit circle is discussed in the matrix setting. An explicit matrix representation of the matrix valued orthogonal polynomials in terms of the moments of the measure is presented. Classical recurrence relations are revisited using the matrix representation of the polynomials. The matrix expressions for the kernel polynomials and the Christoffel–Darboux formulas are presented for the first time.

Keywords

42C99Matrix valued orthogonal polynomialsunit circleSchur complementsrecurrence relationskernel polynomialsChristoffel-Darboux
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 1 , 01 March 2009 , pp. 95 - 104
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Akhiezer, N. I., The classical moment problem and some related questions in analysis. English translation, Hafner Publishing Co, New York, 1965.Google Scholar
[2] Aptekarev, A. I. and Nikishin, E. M., The scattering problem for a discrete Sturm-Liouville operator. Math. Sb. 3(1983), no. 7, 327–358.Google Scholar
[3] Berezanskii, Yu. M., Expansions of eigenfunctions of selfadjoint operators. Transl. Math. Monographs 17, American Mathematical Society, Providence, RI, 1968.Google Scholar
[4] Brezinski, C., The methods of Vorobyev and Lanczos. Linear Algebra Appl. 234(1996), 21–41.Google Scholar
[5] Chihara, T. S., An introduction to orthogonal polynomials. Mathematic and its applications 13, Gordon and Breach Science Publishers, New York, 1978.Google Scholar
[6] Delsarte, P., Genin, Y. V., and Kamp, Y. G., Orthogonal polynomial matrices on the unit circle. IEEE Trans. Circuits and Systems CAS-25(1978), no. 3, 149–160.Google Scholar
[7] Duran, A. J., A generalization of Favard's theorem for polynomials satisfying a recurrence relation. J. Approx. Theory 74(1993), no. 1, 83–109.Google Scholar
[8] Duran, A. J., On orthogonal polynomials with respect to positive definite matrix of measures. Canad. J. Math. 47(1995), no. 1, 88–112.Google Scholar
[9] Duran, A. J., Markov's theorem for orthogonal matrix polynomials. Canad. J. Math. 48(1996), no. 6, 1180–1195.Google Scholar
[10] Duran, A. J., Ratio asymptotics for orthogonal matrix polynomials. J. Approx. Theory 100(1999), no. 2, 304–344.Google Scholar
[11] Duran, A. J., Matrix inner product having a matrix symmetric second order differential operator. Rocky Mountain J. Math. 27(1997), no. 2, 585–600.Google Scholar
[12] Duran, A. J., Functions with given moments and weight functions for orthogonal polynomials. Rocky Mountain J. Math. 23(1993), no. 1, 87–104.Google Scholar
[13] Duran, A. J. and Van Assche, W., Orthogonal matrix polynomials and higher-order recurrence relations. Linear Algebra Appl. 219(1995), 261–280.Google Scholar
[14] Duran, A. J. and Lopez, P., Orthogonal matrix polynomials: zeros and Blumenthal's theorem. J. Approx. Theory 84(1996), no. 1, 96–118.Google Scholar
[15] Duran, A. J. and Polo, B., Gaussian quadrature formulae for matrix weights. Linear Algebra Appl. 355(2002), 119–146.Google Scholar
[16] Freund, R. W. and Malhotra, M., A block-QMR algorithm for non-Hermitian linear systems with multiple right-hand sides. Linear Algebra Appl. 254(1997), 119–157.Google Scholar
[17] Fuhrmann, P. A., Orthogonal matrix polynomials and system theory. Rend. Sem. Mat. Univ. Politec. Torino, 1987, Special Issue, 68–124.Google Scholar
[18] Geronimo, J. S., Scattering theory and matrix orthogonal polynomials on the real line. Circuits Systems Signal Process 1(1982), no. 3–4, 471–495.Google Scholar
[19] Geronimo, J. S., Matrix orthogonal polynomials on the unit circle. J. Math. Phys. 22(1981), no. 7, 1359–1365.Google Scholar
[20] Geronimus, Y. L., Polynomials orthogonal on a circle and their applications. Amer. Math. Soc. Translations 1954, no. 104.Google Scholar
[21] El Guennouni, A., Jbilou, K., and Sadok, H., The block Lanczos method for linear systems with multiple right-hand sides. Appl. Numer. Math. 51(2004), no. 2–3, 243–256.Google Scholar
[22] Golub, G. H. and Van Loan, C. F., Matrix computations. Second edition, Johns Hopkins Series in Mathematical Sciences 3, Johns Hopkins University Press, Baltimore, MD, 1989.Google Scholar
[23] Golub, G. H. and Underwood, R., The block Lanczos methods for computing eigenvalues. In: Mathematical software III, Academic Press, New York, 1977, pp. 364–377.Google Scholar
[24] Krein, M. G., Fundamental aspects of the representation theory of Hermitian operators with deficiency index (m, m), AMS Translations 2, no. 97, Providence, RI, 1971, 75–143.Google Scholar
[25] Krein, M. G., Infinite J-matrices and a matrix-moment problem. Doklady Akad. Nauk SSSR, 69(1949), 125–128.Google Scholar
[26] Lebedev, N. N., Special Functions and their applications. Prentice-Hall, Englewood Cliffs, NJ, 1965.Google Scholar
[27] Marcellán, F. and Gonzàles, I. Rodríguez, A class of matrix orthogonal polynomials on the unit circle. Linear Algebra and Appl. 121(1989), 233–241.Google Scholar
[28] Marcellán, F. and Sansigre, G., On a class of matrix orthogonal polynomials on the real line. Linear Algebra and Appl. 181(1993), 97–109.Google Scholar
[29] Mignolet, M. P., Matrix polynomials orthogonal on the unit circle and accuracy of autoregressive models. J. Comput. Appl. Math. 62(1995), no. 2, 229–238.Google Scholar
[30] Nikishin, E. M., The discrete Sturm-Liouville operator and some problems of function theory. Trudy Sem. Petrovsk. 10(1984) 3–77, 237.Google Scholar
[31] Sinap, A. and Van Assche, W., Orthogonal matrix polynomials and applications. J. Comput. Appl. Math. 66(1996), no. 1–2, 27–52.Google Scholar
[32] Szegö, G., Orthogonal polynomials. Fourth edition, American Mathematical Society, Providence, RI, 1975.Google Scholar