Density results with linear combinations of translates of fundamental solutions

Abstract

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by linear combinations of translates of fundamental solutions of the underlying partial differential operator. The singularities of the fundamental solutions lie outside of over(Ω, -). The domains under consideration may possess holes and they are required to satisfy a rather mild boundary regularity requirement, namely the segment condition. We study approximations with respect to the norms of the spaces Ck (over(Ω, -)) and the spaces of uniformly Hölder continuous functions lipk, σ (over(Ω, -)), and we establish density and non-density results for elliptic operators with constant coefficients. We also provide applications of our density results related to the method of fundamental solutions and to the theory of universal series. © 2008 Elsevier Inc. All rights reserved.