Differential forms: Lagrange interpolation, sampling and approximation on polynomial admissible integral k-meshes

In this work we address the problem of interpolating and approximating differential forms starting from data defined by integration. We show that many aspects of nodal interpolation can naturally be carried to this more general framework; in contrast, some of them require the introduction of geometric and measure theoretic hypotheses. After characterizing the norms of the operators involved, we introduce the concept of admissible integral k-mesh, which allows for the construction of robust approximation schemes, and is 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.