2023/01/26 by Jean B. Lasserre, Lasserre, Jean-Bernard
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Functional Equations Stability Results #Markov Chains and Monte Carlo Methods #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2301.11072
openalex publication_date 2023/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a certain variant (or regularization) Λμn of the standard Christoffel function Λμn associated with a measure μ on a compact set Ω⊂ ℝd. Its reciprocal is now a sum-of-squares polynomial in the variables (x,ε), ε>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 ε>0, and under weak assumptions, limn→∞ ε-dΛμn(ξ,ε)=f(ζε) where f (assumed to be continuous) is the unknown density of μ w.r.t. Lebesgue measure on Ω, and ζε\inB_∞(ξ,ε) (and so f(ζε)≈ f(ξ) when ε>0 is small). This is in contrast with the standard Christoffel function where if limn→∞ ndΛμn(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 Λμn) is just integrating symbolically the monomial basis (xα)α∈ℕdn on the box \x: \Vert x-ξ\Vert_∞<ε/2\, so that 1/Λμn is obtained as an explicit polynomial of (ξ,ε).