A modified Christoffel function and its asymptotic properties - LAAS - Laboratoire d'Analyse et d'Architecture des Systèmes
Journal Articles Journal of Approximation Theory Year : 2023

A modified Christoffel function and its asymptotic properties

Jean-Bernard Lasserre

Abstract

We introduce a certain variant (or regularization) $\tilde{\Lambda}^\mu_n$ of the standard Christoffel function $\Lambda^\mu_n$ associated with a measure $\mu$ on a compact set $\Omega\subset \mathbb{R}^d$. Its reciprocal is now a sum-of-squares polynomial in the variables $(x,\varepsilon)$, $\varepsilon>0$. 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 $\varepsilon>0$, and under weak assumptions, $\lim_{n\to\infty} \varepsilon^{-d}\tilde{\Lambda}^\mu_n(\xi,\varepsilon)=f(\zeta_\varepsilon)$ where $f$ (assumed to be continuous) is the unknown density of $\mu$ w.r.t. Lebesgue measure on $\Omega$, and $\zeta_\varepsilon\in\mathbf{B}_\infty(\xi,\varepsilon)$ (and so $f(\zeta_\varepsilon)\approx f(\xi)$ when $\varepsilon>0$ is small). This is in contrast with the standard Christoffel function where if $\lim_{n\to\infty} n^d\Lambda^\mu_n(\xi)$ exists, it is of the form $f(\xi)/\omega_E(\xi)$ where $\omega_E$ is the density of the equilibrium measure of $\Omega$, usually unknown. At last but not least, the additional computational burden (when compared to computing $\Lambda^\mu_n$) is just integrating symbolically the monomial basis $(x^{\alpha})_{\alpha\in\mathbb{N}^d_n}$ on the box $\{x: \Vert x-\xi\Vert_\infty<\varepsilon/2\}$, so that $1/\tilde{\Lambda}^\mu_n$ is obtained as an explicit polynomial of $(\xi,\varepsilon)$.

Keywords

Fichier principal
Vignette du fichier
extension-jat.pdf (383.02 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-03949201 , version 1 (23-01-2023)
hal-03949201 , version 2 (26-01-2023)

Identifiers

Cite

Jean-Bernard Lasserre. A modified Christoffel function and its asymptotic properties. Journal of Approximation Theory, 2023, 295, pp.105955. ⟨10.1016/j.jat.2023.105955⟩. ⟨hal-03949201v2⟩
56 View
82 Download

Altmetric

Share

More