Relative asymptotics for orthogonal matrix polynomials

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

A connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line

Szego's procedure to connect orthogonal polynomials on the unit circle and orthogonal polynomials on [-1,1] is generalized to nonsymmetric measures. It generates the so-called semi-orthogonal functions on the linear space of Laurent polynomials L, and leads to a new orthogonality structure in the module LxL. This structure can be interpreted in terms of a 2x2 matrix measure on [-1,1], and semi-orthogonal functions provide the corresponding sequence of orthogonal matrix polynomials. This gives a connection between orthogonal polynomials on the unit circle and certain classes of matrix orthogonal polynomials on [-1,1]. As an application, the strong asymptotics of these matrix orthogonal polynomials is derived, obtaining an explicit expression for the corresponding Szego's matrix function.Comment: 28 page

Bilinear semi-classical moment functionals and their integral representation

We introduce the notion of bilinear moment functional and study their general properties. The analogue of Favard's theorem for moment functionals is proven. The notion of semi-classical bilinear functionals is introduced as a generalization of the corresponding notion for moment functionals and motivated by the applications to multi-matrix random models. Integral representations of such functionals are derived and shown to be linearly independent.Comment: 25 pages, 3 figures, minor correction and change to Figure