Interpolating Sequences on Convex Curves in the Open Unit Disc

Abstract

Let $D$ be the open unit disc in the complex plane, and let $C$ be the unit circle. Given a convex curve $\Gamma$ in $D \cup C$, internally tangent to $C$ at one point, then a sequence on $\Gamma$, successive points of which are equally spaced in the hyperbolic (Poincaré) metric, is shown to be interpolating. This result is then applied to the study of the Banach algebra ${H^\infty }$. The Gleason part of a point in the maximal ideal space of ${H^\infty }$ which lies in the closure of a convex curve in $D$ is proved to be nontrivial. In addition, for each point $m$ in the maximal ideal space of ${H^\infty }$ which lies in the closure of a compact subset of $D$ union a point of $C$, an interpolating Blaschke product is constructed whose extension to the maximal ideal space has modulus less than 1 on $m$, and the relevance of this to the Shilov boundary of ${H^\infty }$ is discussed.