A modified Christoffel function and its asymptotic properties
In: ISSN: 0021-9045, 2023
Online
academicJournal
Zugriff:
International audience ; 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 $(\xi,\varepsilon)$.
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A modified Christoffel function and its asymptotic properties
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Autor/in / Beteiligte Person: | Lasserre, Jean-Bernard ; Equipe Polynomial OPtimization (LAAS-POP) ; Laboratoire d'analyse et d'architecture des systèmes (LAAS) ; Université Toulouse Capitole (UT Capitole) ; Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse) ; Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J) ; Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3) ; Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP) ; Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole) ; Université de Toulouse (UT) ; ANR-19-P3IA-0004,ANITI,Artificial and Natural Intelligence Toulouse Institute(2019) ; ANR-20-CE48-0014,NuSCAP,Sûreté numérique pour les preuves assistées par ordinateur(2020) |
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Zeitschrift: | ISSN: 0021-9045, 2023 |
Veröffentlichung: | HAL CCSD ; Elsevier, 2023 |
Medientyp: | academicJournal |
DOI: | 10.1016/j.jat.2023.105955 |
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