Christoffel functions with power type weights

TL;DR

This paper derives precise asymptotic formulas for Christoffel functions with power weights on unions of Jordan curves and arcs, using potential theory and polynomial inverse images to handle different geometric components.

Contribution

It provides new asymptotic results for Christoffel functions on complex supports, employing potential theory and Bessel function discretizations for the first time in this context.

Findings

Asymptotics involve equilibrium measures of the support

Different methods are used for curve and arc components

Potential theory and polynomial inverse images are key tools

Abstract

Precise asymptotics for Christoffel functions are established for power type weights on unions of Jordan curves and arcs. The asymptotics involve the equilibrium measure of the support of the measure. The result at the endpoints of arc components is obtained from the corresponding asymptotics for internal points with respect to a different power weight. On curve components the asymptotic formula is proved via a sharp form of Hilbert's lemniscate theorem while taking polynomial inverse images. The situation is completely different on the arc components, where the local asymptotics is obtained via a discretization of the equilibrium measure with respect to the zeros of an associated Bessel function. The proofs are potential theoretical, and fast decreasing polynomials play an essential role in them.