Strong asymptotics for Christoffel functions of planar measures

TL;DR

This paper establishes strong asymptotic formulas for Christoffel functions associated with a broad class of measures and sets in the complex plane, extending known results to more general and varying weight scenarios.

Contribution

It provides a unified approach to strong asymptotics of Christoffel functions for complex measures, including varying weights and general sets, generalizing previous real-line results.

Findings

Derived strong asymptotics for Christoffel functions in complex plane

Extended known results to measures with varying weights

Included cases with polynomial weights on the real line

Abstract

We prove a version of strong asymptotics of Christoffel functions with varying weights for a general class of sets E and measures in the complex plane. This class includes all regular measures in the sense of Stahl-Totik on regular compact sets E in the plane and even allows varying weights. Our main theorems cover some known results for subsets E of the real line R; in particular, we recover information in the case of E=R with Lebesgue measure dx and weight w(x) = exp(-Q(x)) where Q(x) is a nonnegative, even degree polynomial having positive leading coefficient.