Vol. 74, No. 1, 1978

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

Arne Stray

Vol. 74 (1978), No. 1, 251–255
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 {pn} such that

  1. pn f uniformly on compact subsets of D.
  2. pn 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
Authors
Arne Stray
Mathematics Institute
University of Bergen
5007 Bergen
Norway