Moment estimates of the cloud of a planar measure

TL;DR

This paper introduces a method to compute the moments of a measure's cloud in the plane using Christoffel-Darboux kernels, with error bounds linked to spectral properties of Hankel operators.

Contribution

It provides a novel function theoretic framework and computational scheme for transforming measure moments into cloud moments, emphasizing spectral analysis.

Findings

Transforming measure moments via Christoffel-Darboux kernels

Error bounds depend on Hankel kernel spectral asymptotics

Method applicable to planar measures with compact support

Abstract

With a proper function theoretic definition of the {\it cloud} of a positive measure with compact support in the real plane, a computational scheme of transforming the moments of the original measure into the moments of the uniformly distributed mass on the cloud is described. The main limiting operation involves exclusively truncated Christoffel-Darboux kernels, while error bounds depend on the spectral asymptotics of a Hankel kernel belonging to the Hilbert-Schmidt class.