In this paper necessary and
sufficient conditions on a subset S of the unit disc D are given such that every
bounded analytic function f on D is a pointwise limit of a sequence {fn}n=1∞ of
uniformly continuous analytic functions on D bounded by the sup norm of f
and in addition satisfying sup{|fn(z)|,z ∈,S}≦sup{|f(z)|,z ∈ S} for all
n.