Uniform Approximation by Solutions of Elliptic Equations

Abstract

The space ${H_A}(K)$ of continuous functions on a compact set $K$ in Euclidean space which can be uniformly approximated by solutions of the elliptic, constant-coefficient partial differential equation $Af = 0$ is studied. In particular, it is shown that ${H_A}(K)$ is local, in the same sense as in the theory of rational approximation in the complex plane. Simultaneous approximation of functions and their derivatives is also considered.