Cauchy transform and uniform approximation by polynomial modules

Abstract

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

22 pages. To appear in Journal of Mathematical Analysis and Applications