Pointwise behavior of Christoffel function on planar convex domains

TL;DR

This paper establishes bounds on the Christoffel function for planar convex domains, enabling precise pointwise behavior analysis for specific shapes like superellipses.

Contribution

It introduces a general lower bound based on a modified parallel section function and applies it to compute Christoffel function behavior for certain convex domains.

Findings

Derived a lower bound for Christoffel function on convex domains.

Computed the pointwise behavior of Christoffel function for superelliptical domains.

Provided explicit formulas up to a constant factor for specific convex shapes.

Abstract

We prove a general lower bound on Christoffel function on planar convex domains in terms of a modification of the parallel section function of the domain. For a certain class of planar convex domains, in combination with a recent general upper bound, this allows to compute the pointwise behavior of Christoffel function. We illustrate this approach for the domains $\{(x,y):|x|^\alpha+|y|^\alpha\le1\}$, $1<\alpha<2$, and compute up to a constant factor the required modification of the parallel section function, and, consequently, Christoffel function at an arbitrary interior point of the domain.