Orthogonality measures for orthogonal matrix polynomials with periodic coefficients of recurrence relations

Abstract

where P−1(t) = θ, P0(t) ∈ CN×N , the An are nondegenerate matrices, the Bn are Hermitian matrices, and θ denotes the zero matrix. By the matrix analog of the sequence of Chebyshev polynomials of the second kind [3] we mean the sequence of orthonormal matrix polynomials U n defined by the recurrence relations (1) with An = A, Bn = B, and the initial conditions U −1 (t) = θ, U A,B 0 (t) = I. The measure WA,B(x), with respect to which the polynomials U n are orthonormal was obtained in [1]. In the case of a positive definite matrix A, the measureWA,B(x) has the following form: