Optimal Polynomial Admissible Meshes on Some Classes of Compact Subsets of $\R^d$
This provides a theoretical extension of optimal admissible meshes to broader classes of domains, which is incremental for researchers in approximation theory.
The authors prove that compact subsets of ℝ^d which are closures of bounded star-shaped Lipschitz domains with positive reach admit optimal polynomial admissible meshes, extending a prior result from C^2 to Lipschitz domains. They also constructively show existence of optimal AMs for C^{1,1} domains using a sharp multivariate Bernstein inequality.
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.