Relative asymptotics for orthogonal matrix polynomials

TL;DR

This paper investigates the asymptotic behavior of matrix orthogonal polynomials satisfying non-symmetric recurrence relations, introducing the generalized matrix Nevai class and exploring their ratio asymptotics, explicit forms, and perturbations involving Dirac delta functionals.

Contribution

It introduces the generalized matrix Nevai class, derives ratio asymptotics, and connects matrix orthogonal polynomials with Sobolev inner products and Dirac delta perturbations.

Findings

Derived explicit expressions for generalized matrix Chebyshev polynomials

Established ratio asymptotics within the generalized matrix Nevai class

Linked Dirac delta functionals to discrete Sobolev inner products

Abstract

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.