Cauchy transform and uniform approximation by polynomial modules

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 $U\mathrm{.}$ We show that there exists $\varphi \in A(K,U)$ so that each $f\in A(K,U)$ can uniformly be approximated by $\{p_n+q_n\varphi \}$ on $K$, where $p_n$ and $q_n$ are analytic polynomials in $z$. In particular, $\varphi$ can be chosen as a Cauchy transform of a finite positive measure $\eta$ compactly supported in $\mathbb{C}\setminus U\mathrm{.}$ Recent developments of analytic capacity and Cauchy transform provide us useful tools in our proofs.