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

Abstract

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.

28 pages

Cited by in corpus (4)

• The Analytic Theory of Matrix Orthogonal Polynomials
• Perturbations of Orthogonal Polynomials With Periodic Recursion Coefficients
• Minimal Rank Decoupling of Full-Lattice CMV Operators with Scalar- and Matrix-Valued Verblunsky Coefficients
• Matrix measures on the unit circle, moment spaces, orthogonal polynomials and the Geronimus relations