Sampling, approximation, and interpolation of differential forms by admissible integral k-meshes

Abstract

In this work we introduce the concept of admissible integral $k$-mesh for sampling differential forms with contiuous coefficients on a real body $E\subset {\mathbb{R}}^n$, and provide two techniques for the construction of admissible integral $k$-meshes on real bodies enjoying the Markov or the Bernstein inequality. Admissible integral $k$-meshes allow for the construction of robust approximation schemes, and are used to extract interpolation sets with high stability properties. To this end, the concepts of Fekete currents and Leja sequences of currents are formalized, and a numerical scheme for their approximation is proposed.