Opuscula Math. 36, no. 3 (2016), 409-423
http://dx.doi.org/10.7494/OpMath.2016.36.3.409

 
Opuscula Mathematica

Matrix polynomials orthogonal with respect to a non-symmetric matrix of measures

Marcin J. Zygmunt

Abstract. The paper focuses on matrix-valued polynomials satisfying a three-term recurrence relation with constant matrix coefficients. It is shown that they form an orthogonal system with respect to a matrix of measures, not necessarily symmetric. Moreover, it is stated the condition on the coefficients of the recurrence formula for which the matrix measure is symmetric.

Keywords: matrix orthogonal polynomials, recurrence formula, matrix of measures, block Jacobi matrices.

Mathematics Subject Classification: 47B36, 15A57, 39B42, 42C05.

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  1. Yu.M. Berezanski, Expansions in Eigenfunctions of Selfadjoint Operators, Translations of Mathematical Monographs 17, American Mathematical Society, R.I., 1968.
  2. C. Berg, A.J. Duran, Orthogonal polynomials and analytic functions associated to positive definite matrices, J. Math. Anal. Appl. 315 (2006), 54-67.
  3. T. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and Its Applications 13, Gordon and Breach, New York, 1978.
  4. D. Damanik, A. Pushnitski, B. Simon, The analytic theory of matrix orthogonal polynomials, Surveys in Approx. Theory 4 (2008), 1-85.
  5. H. Dette, B. Reuther, W.J. Studden, M.J. Zygmunt, Matrices measures and random walks with a block tridiagonal transition matrix, SIAM J. Matrix Anal. Appl. 29 (2006) 1, 117-142.
  6. A.J. Duran, A generalization of Favard's theorem for polynomials satisfying a recurrence relation, J. Approx. Theory 74 (1993), 83-109.
  7. A.J. Duran, On orthogonal polynomials with respect to a positive definite matrix of measures, Canad. J. Math. 47 (1995), 88-112.
  8. F.A. Grünbaum, M.D. de la Iglesia, Matrix valued orthogonal polynomials arising from group representation theory and a family of quasi-birth-and-death processes, SIAM J. Matrix Anal. Applic. 30 (2008) 2, 741-761.
  9. M.G. Krein, Infinite \(J\)-matrices and a matrix moment problem, Dokl. Akad. Nauk SSSR 69 (1949), 125-128 [in Russian].
  10. M.A. Naimark, Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of second order on a semi-axis, Trudy Mosk. Mat. Obs. 3 (1954), 181-270 [in Russian].
  11. A. Sinap, W. Van Assche, Orthogonal matrix polynomials and applications, J. Comput. Appl. Math. 66 (1996), 27-52.
  12. G. Szegö, Orthogonal Polynomials, AMS Coll. Pub., vol. 23, AMS, Providence, 1975 (4th edition).
  13. M.J. Zygmunt, Matrix orthogonal polynomials and continued fractions, Linear Alg. Appl. 340, (2002) 1-3, 155-168.
  14. M.J. Zygmunt, Jacobi block matrices with constant matrix terms, Oper. Th.: Adv. & Appl. 154 (2004), 233-238.
  15. M.J. Zygmunt, Non symmetric random walk on infinite graph, Opuscula Math. 31, (2011) 4, 669-674.
  • Marcin J. Zygmunt
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Krakow, Poland
  • Communicated by P.A. Cojuhari.
  • Received: 2015-03-08.
  • Revised: 2015-11-25.
  • Accepted: 2015-11-28.
  • Published online: 2016-02-21.
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Cite this article as:
Marcin J. Zygmunt, Matrix polynomials orthogonal with respect to a non-symmetric matrix of measures, Opuscula Math. 36, no. 3 (2016), 409-423, http://dx.doi.org/10.7494/OpMath.2016.36.3.409

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