Approximation by simple partial fractions with constraints on the poles. II

It is shown that if a compact set $K$ not separating the plane $\mathbb C$ lies in the union $\widehat{E}\setminus E$ of the bounded components of the complement of another compact set $E$, then the simple partial fractions (the logarithmic derivatives of polynomials) with poles in $E$ are dense in the space $AC(K)$ of functions that are continuous on $K$ and analytic in its interior. It is also shown that if a compact set $K$ with connected complement lies in the complement $\mathbb C\setminus\overline{D}$ of the closure of a doubly connected domain $D\subset \overline{\mathbb C}$ with bounded connected components of the boundary $E^+$ and $E^-$, then the differences $r_1-r_2$ of the simple partial fractions such that $r_1$ has its poles in $E^+$ and $r_2$ has its poles in $E^-$ are dense in the space $AC(K)$.
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