Orthogonal Matrix Polynomials, Connection Between Recurrences on the Unit Circle and on a Finite Interval

Perturbations of orthogonal polynomials with periodic recursion coefficients

“…The shift from type 2 to 3 is easy on account of Proposition 6.4. By applying the argument of [118], we getà n → 1 for the equivalentJ of type 2, conclude the whole equivalence class is in the Nevai class, and see A n → 1. So we will only worry about the changes needed to go from σ(J) = [−2, 2] to σ ess (J) = [−2, 2], where we follow Denisov's approach for the scalar case [31].…”

Large deviations and a new sum rule for spectral matrix measures of the Jacobi ensemble

“…In this Section, we write Σ R for a matrix measure on the real line and denote by Σ T =Sz −1 (Σ R ) the preimage under the Szegő mapping. The correspondence between polynomials orthogonal with respect to Σ T and with respect to Σ R is ruled by the following theorem (see Proposition 1 in [37]). It is the matrix version of a famous theorem due to Szegő [29].…”

Matrix measures on the unit circle, moment spaces, orthogonal polynomials and the Geronimus relations

“…In this section we present a new proof of the Geronimus relations, which provide a representation of the canonical moments (or Verblunsky coefficients) of a symmetric matrix measure on the unit circle in terms of the coefficients in the recurrence relations of a sequence of orthogonal polynomials with respect to an associated matrix measure on the interval [−1, 1]. There exists several alternative proofs of these relations in the literature [see Yakhlef and Marcellán (2001) and Damanik et al (2008)], but the one presented here explicitly uses the theory of canonical moments of matrix measures as introduced in Dette and Studden (2002). As a by-product we derive several interesting properties of the Verblunsky coefficients.…”