Asymptotic behaviour of Christoffel-Darboux kernel via three-term recurrence relation I

TL;DR

This paper investigates the asymptotic behavior of Christoffel functions and the Christoffel-Darboux kernel for Jacobi parameters in specific classes, revealing their scaling limits under regularity conditions.

Contribution

It provides new asymptotic results for Christoffel functions and kernels for Jacobi parameters in asymptotically periodic, periodically modulated, and mixed classes.

Findings

Asymptotic behavior characterized for three classes of Jacobi parameters.

Scaling limits of Christoffel-Darboux kernel established.

Results applicable to measures with compact supports.

Abstract

For Jacobi parameters belonging to one of the three classes: asymptotically periodic, periodically modulated and the blend of these two, we study the asymptotic behavior of the Christoffel functions and the scaling limits of the Christoffel-Darboux kernel. We assume regularity of Jacobi parameters in terms of the Stolz class. We emphasize that the first class only gives rise to measures with compact supports.