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

• Authors: M. J. Cantero;M. P. Ferrer;L. Moral;L. Velázquez
• Venue: Journal of Computational and Applied Mathematics
• Year: 2003

Szegö'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 Szegö's matrix function.