A modified Christoffel function and its asymptotic properties

Abstract

We introduce a certain variant (or regularization) Λ̃nμ of the standard Christoffel function Λnμ associated with a measure μ on a compact set Ω⊂Rd. Its reciprocal is now a sum-of-squares polynomial in the variables (x,ɛ), ɛ>0. It shares the same dichotomy property of the standard Christoffel function, that is, the growth with n of its inverse is at most polynomial inside and exponential outside the support of the measure. Its distinguishing and crucial feature states that for fixed ɛ>0, and under weak assumptions, limn→∞ɛ−dΛ̃nμ(ξ,ɛ)=f(ζɛ) where f (assumed to be continuous) is the unknown density of μ w.r.t. Lebesgue measure on Ω, and ζɛ∈B∞(ξ,ɛ) (and so f(ζɛ)≈f(ξ) when ɛ>0 is small). This is in contrast with the standard Christoffel function where if limn→∞ndΛnμ(ξ) exists, it is of the form f(ξ)/ωE(ξ) where ωE is the density of the equilibrium measure of Ω, usually unknown. At last but not least, the additional computational burden (when compared to computing Λnμ) is just integrating symbolically the monomial basis (xα)α∈Nnd on the box {x:‖x−ξ‖∞<ɛ/2}, so that 1/Λ̃nμ is obtained as an explicit polynomial of (ξ,ɛ).