Cauchy transform and uniform approximation by polynomial modules

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