A scheme for approximating bounded analytic functions on certain subsets of the unit disc

Abstract

We denote by H the space of all bounded analytic functions in the unit disc D = {z: \z\ > 1}. We consider a relatively closed subset S of with the following property: If /e 0 and its restriction f\ s to S is uniformly continuous, there exists a bounded sequence of polynomials {p n } such that (i) Pn^f uniformly on compact subsets of D. (ii) p n ~>/uniformly on S. A set S with this property, is called a Mergelyan set for H. The object of this paper is to give a new and constructive proof of the following result: THEOREM. Let S be a relatively closed subset of D. Assume that the clusterpoints of S on the unit circle T -{z: \z\ = 1} which are not in the nontangential closure of S 9 has zero linear measure. Then S is a Mergelyan set for H.