Representation of continuous functions as sums of Green functions

Let $K\subset \Omega\subseteq \mathbb{R}^n$, where $K$ is polar and compact and $\Omega$ is a domain with Green function $G_\Omega({\boldsymbol\cdot},{\boldsymbol\cdot} )$. We characterize those subsets $E$ of $\Omega\backslash K$ which have the following property: Every positive continuous function on $K$ can be written as $\sum_k\lambda_kG_\Omega(x_k, {\boldsymbol\cdot})$, where $x_k\in E$ and $\lambda_k>0$ for each $k$.