Abstract
In ?2, we use this theorem to establish an analogous result in the setting of finite open Riemann surfaces. ??3 and 4 consider certain questions which arise naturally in the course of the proof of this generalization. We mention that the chief result of ?2, Theorem 2.6, has been obtained independently by N. L. Alling [3] who has used methods more highly algebraic than ours. The second matter we shall be concerned with is that of interpolation. If R is a Riemann surface and if E is a subset of R, call E an interpolation set for R if for every bounded complex-valued function a on E, there is a bounded holomorphic function f on R such that f| E = . Carleson [6] has characterized interpolation sets in the unit disc:
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