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
- pn → f uniformly on compact subsets of D.
- 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∞.
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