Christoffel functions for multiple orthogonal polynomials

TL;DR

This paper investigates the asymptotic behavior of Christoffel--Darboux kernels for multiple orthogonal polynomials, revealing their connection to zero distribution and extending Nevai's operators within this framework.

Contribution

It provides new insights into the weak asymptotics of the Christoffel--Darboux kernel and extends Nevai's operators to multiple orthogonal polynomials.

Findings

Weak limit of the scaled Christoffel--Darboux kernel matches the zero counting measure limit.

Established conditions under which the asymptotic behavior holds.

Extended Nevai's operators to the context of multiple orthogonal polynomials.

Abstract

We study weak asymptotic behaviour of the Christoffel--Darboux kernel on the main diagonal corresponding to multiple orthogonal polynomials. We show that under some hypotheses the weak limit of $\tfrac{1}{n} K_n(x,x)\, d\mu$ is the same as the limit of the normalized zero counting measure of type II multiple orthogonal polynomials. We also study an extension of Nevai's operators to our context.