Relative asymptotics for orthogonal matrix polynomials

• Published in: Linear algebra and its applications Vol. 437; no. 7; pp. 1458 - 1481
• Main Authors: Branquinho, A., Marcellán, F., Mendes, A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.10.2012; Elsevier
• Subjects: Algebra; Asymptotic results; Exact sciences and technology; Linear and multilinear algebra, matrix theory; Linear functional; Mathematical analysis; Mathematics; Matrix orthogonal polynomials; Nevai class; Recurrence relation; Sciences and techniques of general use; Special functions; Tridiagonal operator; 33C45; Asymptotic results; 39B42; Matrix orthogonal polynomials; Recurrence relation; Tridiagonal operator; Nevai class; Linear functional; Orthogonal matrix; Polynomial matrix; Chebyshev polynomial; Functional equation; Orthogonality; Modeling; Orthogonal polynomial; Hypergeometric function; Matrix polynomial; Asymptotic approximation
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2012.04.032

In this paper we study sequences of matrix polynomials that satisfy a non-symmetric recurrence relation. To study this kind of sequences we use a vector interpretation of the matrix orthogonality. In the context of these sequences of matrix polynomials we introduce the concept of the generalized matrix Nevai class and we give the ratio asymptotics between two consecutive polynomials belonging to this class. We study the generalized matrix Chebyshev polynomials and we deduce its explicit expression as well as we show some illustrative examples. The concept of a Dirac delta functional is introduced. We show how the vector model that includes a Dirac delta functional is a representation of a discrete Sobolev inner product. It also allows to reinterpret such perturbations in the usual matrix Nevai class. Finally, the relative asymptotics between a polynomial in the generalized matrix Nevai class and a polynomial that is orthogonal to a modification of the corresponding matrix measure by the addition of a Dirac delta functional is deduced.