Cauchy transform and uniform approximation by polynomial modules

Abstract

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⊂K, let A(K,U)⊂C(K) be the algebra of functions that are analytic in U. We show that there exists ϕ∈A(K,U) so that each f∈A(K,U) can uniformly be approximated by {pn+qnϕ} on K, where pn and qn are analytic polynomials in z. In particular, ϕ can be chosen as a Cauchy transform of a finite positive measure η compactly supported in C∖U. Recent developments of analytic capacity and Cauchy transform provide us useful tools in our proofs.