An interpolation problem for subalgebras of<i>H</i>^∞

Abstract

Let E be a closed subset of the unit circle C = {z: \ z \ = 1} and denote by B E the algebra of all functions which are bounded and continuous on the set X = {z: | z | ^ 1, zg E} and analytic in the open disc D = {z:\z\ < 1}. An interpolation set for B E is a relatively closed subset S of X with the property that if a is a bounded and continuous function on (all functions are complex-valued), there is a function / in B E such that f(z) = a(z) for every zeS. The main result of the paper characterizes the interpolation sets for B E as those sets S for which S D is an interpolation set for iJ and S n (C -E) has Lebesgue measure 0. If, in addition, S n D = then S is a peak interpolation set for B E . Also, through a construction process inspired by recent work of J. F. 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.