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,z∉E} 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.
