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

Szegőʹs 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 Szegőʹs matrix function.