Vol. 28, No. 3, 1969

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ISSN: 0030-8730
An interpolation problem for subalgebras of H∞

E. A. Heard and James Howard Wells

Vol. 28 (1969), No. 3, 543–553

Let E be a closed subset of the unit circle C = {z : |z| = 1} and denote by BE the algebra of all functions which are bounded and continuous on the set X = {z : |z|1,zE} and analytic in the open disc D = {z : |z| < 1}. An interpolation set for BE is a relatively closed subset S of X with the property that if α is a bounded and continuous function on S (all functions are complex-valued), there is a function f in BE such that f(z) = α(z) for every z S. The main result of the paper characterizes the interpolation sets for BE as those sets S for which S D is an interpolation set for H and S (C E) has Lebesgue measure 0. If, in addition, S D = ϕ then S is a peak interpolation set for BE. Also, through a construction process inspired by recent work of J. P. Kahane, it is shown that the existence of peak points for a sup norm algebra of continuous functions on a compact, connected space implies the existence of infinite interpolation sets relative to the algebra and certain of its weak extensions.

Mathematical Subject Classification
Primary: 46.55
Secondary: 30.00
Received: 8 January 1968
Published: 1 March 1969
E. A. Heard
James Howard Wells