Cauchy transform and uniform approximation by polynomial modules

Abstract: For a compact subset

K

of the complex plane

m a t h b b C,

let

C (K)

denote the algebra of continuous functions on

K

. For an open subset

U s u b s e t K,

let

A (K, U) s u b s e t C (K)

be the algebra of functions that are analytic in

U .

We show that there exists

p h i i n A (K, U)

so that each

f i n A (K, U)

can uniformly be approximated by

p_{n} + q_{n} p h i

on

K

, where

p_{n}

and

q_{n}

are analytic polynomials in

z

. In particular,

p h i

can be chosen as a Cauchy transform of a finite positive measure

e t a

compactly supported in

m a t h b b C s e t m i n u s U .

Recent developments of analytic capacity and Cauchy transform provide us useful tools in our proofs.

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