Representation of continuous functions as sums of Green functions

Gardiner, Stephen

doi:10.1090/S0002-9939-96-03176-0

Representation of continuous functions as sums of Green functions
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by Stephen J. Gardiner PDF

Proc. Amer. Math. Soc. 124 (1996), 1149-1157 Request permission

Abstract:

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 $k$.

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