Density results with linear combinations of translates of fundamental solutions

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain ΩΩ by linear combinations of translates of fundamental solutions of the underlying partial differential operator. The singularities of the fundamental solutions lie outside of 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(Ω¯) and the spaces of uniformly Hölder continuous functions lipk,σ(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.