Approximation and interpolation

Arne Stray

Vol. 40 (1972), No. 2, 463–475

DOI: 10.2140/pjm.1972.40.463

Abstract

Let X be a compact plane set, X⁰ its interior, and suppose E is a subset of ∂X = X∖X⁰.H^∞(X⁰) is the algebra of all bounded analytic functions on X⁰ and H_E^∞(X⁰) denotes all bounded continuous functions on X⁰ ∪ E analytic in X⁰.

Interpolation sets for H_F^∞(X⁰) are studied if E is open relative to ∂X.

If X satisfies certain conditions which involve analytic capacity, it is shown that an interpolation set S for H^∞(X⁰) is an interpolation set for H^∞(0) for some open set 0 which contains every point of X except the points on ∂X in the closure of S. Similar results are proved for R(X) without restrictions on X.

Mathematical Subject Classification

Primary: 30A82

Milestones

Received: 9 December 1970

Published: 1 February 1972