Let X be a compact plane set, X° its interior, and suppose E is a subset of dX = X\X°. H°°(X 0 ) is the algebra of all bounded analytic functions on X° and H£(X°) denotes all bounded continuous functions on X° u E analytic in X°.Interpolation sets for HE(X°) are studied if E is open relative to dX.If X satisfies certain conditions which involve analytic capacity, it is shown that an interpolation set S for H°°(X°) is an interpolation set for ϋ°°(0) for some open set 0 which contains every point of X except the points on dX in the closure of S. Similar results are proved for R(X) without restrictions on X.Introduction and notation* The paper is divided into three

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Necessary and sufficient conditions on a relatively closed subset F F of D = { z : | z | > 1 } D = \{ z:|z| > 1\} are given such that each analytic function in D D which is uniformly continuous on F F can be uniformly approximated by polynomials on K ∪ F K \cup F for each compact subset K K of D D .