Stability of Asymptotics of Christoffel-Darboux Kernels

TL;DR

This paper investigates how the convergence of Christoffel-Darboux kernels to the sine kernel remains stable under various perturbations of the underlying measure's Jacobi coefficients, and explores implications for measure singularity.

Contribution

It establishes stability results for the sine kernel convergence under boundary, $ ext{l}^1$, and random $ ext{l}^2$ perturbations of Jacobi coefficients, and links convergence to measure properties.

Findings

Stability under boundary condition variations.

Weak stability under $ ext{l}^1$ and random $ ext{l}^2$ perturbations.

Convergence to sine kernel implies measure has no point masses.

Abstract

We study the stability of convergence of the Christoffel-Darboux kernel, associated with a compactly supported measure, to the sine kernel, under perturbations of the Jacobi coefficients of the measure. We prove stability under variations of the boundary conditions and stability in a weak sense under $\ell^1$ and random $\ell^2$ diagonal perturbations. We also show that convergence to the sine kernel at $x$ implies that $\mu(\{x\})=0$.