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
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.
- Yu.M. Berezanski, Expansions in Eigenfunctions of Selfadjoint Operators, Translations of Mathematical Monographs 17, American Mathematical Society, R.I., 1968.
- C. Berg, A.J. Duran, Orthogonal polynomials and analytic functions associated to positive definite matrices, J. Math. Anal. Appl. 315 (2006), 54-67.
- T. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and Its Applications 13, Gordon and Breach, New York, 1978.
- D. Damanik, A. Pushnitski, B. Simon, The analytic theory of matrix orthogonal polynomials, Surveys in Approx. Theory 4 (2008), 1-85.
- 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.
- A.J. Duran, A generalization of Favard's theorem for polynomials satisfying a recurrence relation, J. Approx. Theory 74 (1993), 83-109.
- A.J. Duran, On orthogonal polynomials with respect to a positive definite matrix of measures, Canad. J. Math. 47 (1995), 88-112.
- 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.
- M.G. Krein, Infinite \(J\)-matrices and a matrix moment problem, Dokl. Akad. Nauk SSSR 69 (1949), 125-128 [in Russian].
- 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].
- A. Sinap, W. Van Assche, Orthogonal matrix polynomials and applications, J. Comput. Appl. Math. 66 (1996), 27-52.
- G. Szegö, Orthogonal Polynomials, AMS Coll. Pub., vol. 23, AMS, Providence, 1975 (4th edition).
- M.J. Zygmunt, Matrix orthogonal polynomials and continued fractions, Linear Alg. Appl. 340, (2002) 1-3, 155-168.
- M.J. Zygmunt, Jacobi block matrices with constant matrix terms, Oper. Th.: Adv. & Appl. 154 (2004), 233-238.
- 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.
