Uniform approximability of functions by polynomial solutions of second-order elliptic equations on compact plane sets

We investigate conditions for the uniform approximability of functions by polynomial solutions of second-order elliptic equations with constant complex coefficients on compact sets in $\mathbb R^2$. Some new results of a reductive nature are obtained which ensure that a compact set is an approximation compactum if certain special subsets with a simpler topological structure have this property.