Boundary measures of analytic differentials and uniform approximation on a Riemann surface
Laura Kodama · Pacific Journal of Mathematics · 1965
A classical theorem of F. and M β Riesz establishes a oneto-one correspondence between analytic differentials of class Hi on the interior of the unit disc and finite complex-valued Borel measures on the boundary of the disc which are orthogonal to polynomials.The main result of this paper gives a similar correspondence when the unit disc is replaced by a compact subset, satisfying a finite connectivity condition, of any noncompact Riemann surface.The analytic differentials on the interior of the set satisfy a boundedness condition analogous to the classical H λ differentials and the measures on the boundary of the set are those orthogonal to all meromorphic functions with a finite number of poles in the complement of the set.This result is then used to obtain theorems on uniform approximation on the set by such meromorphic functions.This paper extends results of Bishop in [2] and [5] where he considers compact subsets of the plane staisf ying a simple connectivity condition.1 He obtained such a one-to-one correspondence between boundary measures and analytic differentials and used his result together with an approximation theorem for nowhere dense sets to give a proof of Mergelyan's approximation theorem [6].We are able to extend Mergelyan's theorem to our more general sets and also show that "local" approximation implies approximation on the whole set. I* Boundary measures of analytic differentials* A. DEFINITIONS AND PRELIMINARIES.In this section S will denote an open Riemann surface.If K is a compact subset of S, we denote by C(K) the algebra of all continuous complex-valued functions on K with norm ||/|| = sup \f(x) |, and by xEK A(K) the closed subalgebra of C(K) consisting of those functions which are limits of meromorphic functions on S with finitely many poles in