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

Arne Stray

Vol. 74 (1978), No. 1, 251–255

DOI: 10.2140/pjm.1978.74.251

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 D with the following property: If f ∈ H^∞ and its restriction f|_S to S is uniformly continuous, there exists a bounded sequence of polynomials {p_n} such that

1. p_n → f uniformly on compact subsets of D.
2. p_n → f 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, has zero linear measure. Then S is a Mergelyan set for H^∞.

Mathematical Subject Classification

Primary: 30A82, 30A82

Milestones

Received: 18 January 1977

Revised: 24 August 1977

Published: 1 January 1978