A disintegration of the Christoffel function

TL;DR

This paper demonstrates that the Christoffel function can be factorized into marginal and conditional components, revealing new insights into its structure and connecting polynomial optimization with orthogonal polynomials.

Contribution

It introduces a novel factorization of the Christoffel function into marginal and conditional parts, and highlights an overlooked property linking sum-of-squares polynomials to Christoffel functions.

Findings

Christoffel function factorizes into marginal and conditional CFs

Sum-of-squares polynomials are Christoffel functions of some linear form

Establishes a connection between polynomial optimization and orthogonal polynomials

Abstract

We show that the Christoffel function (CF) factorizes (or can be disintegrated) as the product of two Christoffel functions, one associated with the marginal and the another related to the conditional distribution, in the spirit of "the CF of the disintegration is the disintegration of the CFs". In the proof one uses an apparently overlooked property (but interesting in its own) which states that any sum-of-squares polynomial is the Christoffel function of some linear form (with a representing measure in the univariate case). The same is true for the convex cone of polynomials that are positive on a basic semi-algebraic set. This interpretation of the CF establishes another bridge between polynomials optimization and orthogonal polynomials.