Ratio asymptotics for orthogonal matrix polynomials
type: Item
instance of: scholarly article
title: Ratio asymptotics for orthogonal matrix polynomials
zbMATH Open document ID: 0944.42015
author: A. J. Durán
published in: Journal of Approximation Theory
publication date: 2000-09-17T00:00:00Z
review text: In the 1979's monograph ``Orthogonal polynomials'' [Mem. Am. Math. Soc. 213, 185 p. (1979; Zbl 0405.33009)] \textit{P. Nevai} established a remarkable result that orthogonal polynomials on \(\mathbb R\), satisfying a 3-term recurrence relation with converging coefficients, present a ratio asymptotics outside of the set of accumulation points of the zeros of these polynomials. The limit of the ratio of consecutive polynomials is the Cauchy transform of the weight of orthogonality for Chebyshev polynomials of the second kind. The present paper extends these results to orthogonal matrix polynomials. Namely, if for \(n \geq 0\), \(A_n\) are \(N \times N\) nonsingular matrices and \(B_n\) are \(N \times N\) Hermitian matrices, the recurrence relation \[ t P_n(t)=A_{n+1} P_{n+1}(t) + B_n P_n(t) +A_n^* P_{n-1}(t) , \quad P_{-1}=0 , \quad P_0=I , \] is considered under the assumption that \[ \lim_n A_n =A , \quad \lim_n B_n =B ; \] (in analogy with the scalar case, this setting defines the Nevai class \(M(A,B)\) of matrix polynomials or matrix measures of orthogonality). In the case when \(A\) is nonsingular, it is shown that the sequence \(P_{n-1}(z) P_n^{-1}(z)A_n^{-1}\) converges to a matrix-valued function \(F_{A,B}(z)\) outside of the set \(\Gamma\) of accumulation points of zeros of \(\det (P_n)\). The function \(F_{A,B}\) is in fact the Cauchy transform of the positive definite matrix of measures \(W_{A,B}\) which generalizes the orthogonality weight for Chebyshev polynomials of the second kind. The author finds explicit expressions for \(F_{A,B}\) (when \(A\) is Hermitian) and for \(W_{A,B}\) (when \(A\) is positive definite). Nevertheless, if \(A\) is just nonsingular, many questions about both \(F_{A,B}\) and \(W_{A,B}\) remain open. The last section of the paper is devoted to the degenerate case, that is, when \(A\) is singular. Then, the sequence \(P_{n-1}(z) P_n^{-1}(z)A_n^{-1}\) still converges to \(F_{A,B}(z)\) in \(\mathbb C \setminus \Gamma\), and \(F_{A,B}…
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 30E15
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 1375227
zbMATH Keywords: matrix orthogonal polynomials
zbMATH Keywords: asymptotics
zbMATH Keywords: Nevai class
zbMATH Keywords: recurrence relation
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
full work available at URL: jath.1999.3351
OpenAlex ID: W2069821071
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DOI: 10.1006/JATH.1999.3351
description: scientific article; zbMATH DE number 1375227