Approximation properties of harmonic vector fields and differential forms

Abstract

This work is an attempt to find a multidimensional generalization of the planar theory of uniform rational approximation. Harmonic differential forms in ℝⁿ are considered as analogues of analytic functions in ℂ, whereas rational functions of a complex variable are replaced by the so-called Biot-Savard forms (with singularities on appropriate cycles instead of points). A Runge-like theorem is proved. Theorems by Hartogs-Rosenthal and Rao (on approximation by harmonic gradients in ℝ³) are generalized to any dimension and any degree of forms.