Cauchy transform and uniform approximation by polynomial modules

TL;DR

This paper demonstrates that functions analytic on a subset of a compact set can be uniformly approximated by polynomial modules involving a Cauchy transform, leveraging recent advances in analytic capacity and Cauchy transform theory.

Contribution

It introduces a method to approximate functions in $A(K,U)$ using polynomial modules with a Cauchy transform, extending approximation techniques in complex analysis.

Findings

Existence of a specific $$ in $A(K,U)$ for approximation.

Approximation of functions by polynomial modules with $$.

Use of analytic capacity and Cauchy transform in proofs.

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 $\phi\in A(K,U)$ 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.