2 00 2 A connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line

Szeg˝ o'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 Szeg˝ o's matrix function.