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

Abstract

Szegő'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 Λ, and leads to a new orthogonality structure in the module Λ× Λ. This structure can be interpreted in terms of a 2×2 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 Szegő's matrix function.