A modified Christoffel function and its asymptotic properties

Abstract

We introduce a certain variant (or regularization) ${\tilde{\Lambda }}_n^{\mu }$ of the standard Christoffel function ${\Lambda }_n^{\mu }$ associated with a measure $\mu$ on a compact set $\Omega \subset R^d$. 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, ${lim}_{n\to \infty }{\varepsilon }^{-d}{\tilde{\Lambda }}_n^{\mu }(\xi ,\varepsilon )=f\left({\zeta }_{\varepsilon }\right)$ where $f$ (assumed to be continuous) is the unknown density of $\mu$ w.r.t. Lebesgue measure on $\Omega$, and ${\zeta }_{\varepsilon }\in B_{\infty }(\xi ,\varepsilon )$ (and so $f\left({\zeta }_{\varepsilon }\right)\approx f\left(\xi \right)$ when $\varepsilon >0$ is small). This is in contrast with the standard Christoffel function where if ${lim}_{n\to \infty }{n}^d{\Lambda }_n^{\mu }\left(\xi \right)$ exists, it is of the form $f\left(\xi \right)/{\omega }_E\left(\xi \right)$ 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 ${\Lambda }_n^{\mu }$) is just integrating symbolically the monomial basis ${\left(x^{\alpha }\right)}_{\alpha \in N_n^d}$ on the box $\{x:{\Vert x-\xi \Vert }_{\infty }<\varepsilon /2\}$, so that $1/{\tilde{\Lambda }}_n^{\mu }$ is obtained as an explicit polynomial of $(\xi ,\varepsilon )$.