Federico Piazzon, Alvise Sommariva, Marco Vianello
We present a brief survey on the compression of discrete measures by Caratheodory-Tchakaloff Subsampling, its implementation by Linear or Quadratic Programming and the application to multivariate polynomial Least Squares. We also give an algorithm that computes the corresponding Caratheodory-Tchakaloff (CATCH) points and weights for polynomial spaces on compact sets and manifolds in 2D and 3D.
Federico Piazzon
We introduce numerical methods for the approximation of the main (global) quantities in Pluripotential Theory as the \emph{extremal plurisubharmonic function} $V_E^*$ of a compact $\mathcal L$-regular set $E\subset \C^n$, its \emph{transfinite diameter} $δ(E),$ and the \emph{pluripotential equilibrium measure} $μ_E:=\ddcn{V_E^*}.$ The methods rely on the computation of a \emph{polynomial mesh} for $E$ and numerical orthonormalization of a suitable basis of polynomials. We prove the convergence of the approximation of $δ(E)$ and the uniform convergence of our approximation to $V_E^*$ on all $\C^n;$ the convergence of the proposed approximation to $μ_E$ follows. Our algorithms are based on the properties of polynomial meshes and Bernstein Markov measures. Numerical tests are presented for some simple cases with $E\subset \R^2$ to illustrate the performances of the proposed methods.
Federico Piazzon
We show that any compact subset of $\R^d$ which is the closure of a bounded star-shaped Lipschitz domain $Ω$, such that $\complement Ω$ has positive reach in the sense of Federer, admits an \emph{optimal AM} (admissible mesh), that is a sequence of polynomial norming sets with optimal cardinality. This extends a recent result of A. Kroó on $\mathscr C^ 2$ star-shaped domains. Moreover, we prove constructively the existence of an optimal AM for any $K := \overlineΩ\subset \R^ d$ where $Ω$ is a bounded $\mathscr C^{ 1,1}$ domain. This is done by a particular multivariate sharp version of the Bernstein Inequality via the distance function.
Federico Piazzon
The Bernstein Markov Property, shortly BMP, is an asymptotic quan- titative assumption on the growth of uniform norms of polynomials or rational functions on a compact set with respect to L μ 2 -norms, where μ is a positive finite measure. We consider two variants of BMP for rational functions with restricted poles and compare them with the polynomial BMP finding out some sufficient condi- tions for the latter to imply the former. Moreover, we recover a sufficient mass- density condition for a measure to satisfy the rational BMP on its support.