Asymptotic Behaviour of Christoffel–Darboux Kernel Via Three-Term Recurrence Relation I

Abstract

For Jacobi parameters belonging to one of 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.

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

“…The case when (4.29) is violated is more complicated and demands stronger hypotheses. We refer to [23] for more details.…”

Orthogonal Polynomials with Periodically Modulated Recurrence Coefficients in the Jordan Block Case II

“…For unbounded case, see e.g. [10,16,21,39,42,[57][58][59][60] and the references therein. In this article we consider unbounded Jacobi matrices only.…”

“…Let us remark that the first order asymptotics of generalized eigenvectors provided by Theorem C is insufficient to prove (1.13). It is an open problem whether, similarly to [57][58][59]61], one can relate the value of (1.13) to the density of the measure μ. We hope to return to this problem in the future.…”