A criterion for uniform approximability on arbitrary compact sets for solutions of elliptic equations

Let $X$ be an arbitrary compact subset of the plane. It is proved that if $L$ is a homogeneous elliptic operator with constant coefficients and locally bounded fundamental solution, then each function $f$ that is continuous on $X$ and satisfies the equation $Lf=0$ at all interior points of $X$ can be uniformly approximated on $X$ by solutions of the same equation having singularities outside $X$. A theorem on uniform piecemeal approximation of a function is also established under weaker constraints than in the standard Vitushkin scheme.
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