Constructive quantization: approximation by empirical measures

In this article, we study the approximation of a probability measure $\mu$ on $\mathbb{R}^{d}$ by its empirical measure $\hat{\mu}_{N}$ interpreted as a random quantization. As error criterion we consider an averaged $p$-th moment Wasserstein metric. In the case where $2p<d$, we establish refined upper and lower bounds for the error, a high-resolution formula. Moreover, we provide a universal estimate based on moments, a so-called Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.Comment: 22 page

Constructive approximation in de Branges-Rovnyak spaces

In most classical holomorphic function spaces on the unit disk, a function $f$ can be approximated in the norm of the space by its dilates f\_r(z):=f(rz)~(r \textless{} 1). We show that this is \emph{not} the case for the de Branges--Rovnyak spaces \cH(b). More precisely, we give an example of a non-extreme point $b$ of the unit ball of $H^\infty$ and a function f\in\cH(b) such that \lim\_{r\to1^-}\|f\_r\|\_{\cH(b)}=\infty. It is known that, if $b$ is a non-extreme point of the unit ball of $H^\infty$, then polynomials are dense in \cH(b). We give the first constructive proof of this fact