  For a compact subset $K$ of the complex plane $\mathbb C,$ let $C(K)$ denote
the algebra of continuous functions on $K$. For an open subset $U \subset K,$
let $A(K,U) \subset C(K)$ be the algebra of functions that are analytic in $U.$
We show that there exists $\phi\in A(K,U)$ so that each $f\in A(K,U)$ can
uniformly be approximated by $\{p_n + q_n\phi\}$ on $K$, where $p_n$ and $q_n$
are analytic polynomials in $z$. In particular, $\phi$ can be chosen as a
Cauchy transform of a finite positive measure $\eta$ compactly supported in
$\mathbb C \setminus U.$ Recent developments of analytic capacity and Cauchy
transform provide us useful tools in our proofs.
