Szego asymptotics for matrix-valued measures with countably many bound states

TL;DR

This paper extends Szego asymptotics to matrix-valued measures with infinitely many bound states, providing a detailed analysis of orthonormal polynomials' behavior under specific spectral conditions.

Contribution

It generalizes previous finite-mass results to countably infinite bound states, under Szego and Blaschke-type conditions, for matrix-valued measures.

Findings

Asymptotic behavior of orthonormal polynomials established

Generalization from finite to countably infinite mass points

Extension of scalar and matrix-valued Szego asymptotics results

Abstract

Let $\mu$ be a matrix-valued measure with the essential spectrum a single interval and countably many point masses outside of it. Under the assumption that the absolutely continuous part of $\mu$ satisfies Szego's condition and the point masses satisfy a Blaschke-type condition, we obtain the asymptotic behavior of the orthonormal polynomials on and off the support of the measure. The result generalizes the scalar analogue of Peherstorfer-Yuditskii and the matrix-valued result of Aptekarev-Nikishin, which handles only a finite number of mass points.