Vol. 107, No. 1, 1983

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Gauthier’s localization theorem on meromorphic uniform approximation

Stephen Scheinberg

Vol. 107 (1983), No. 1, 223–228

This paper provides a proof of the localization theorem of Gauthier, which states that a function on a closed subset which is essentially of finite genus in an open Riemann surface is uniformly approximable by global meromorphic functions if and only if it is uniformly approximable locally by local meromorphic functions. The proof relies upon previously published work of Gauthier and of the author and these two lemmas: a connected surface of infinite genus cannot be the union of a compact set and a collection of pair-wise disjoint open sets of finite genus; if a Laurent series for an isolated essential singularity is prescribed at a point of a compact Riemann surface, there can be found an analytic function on the surface with a singularity only at the given point and with Laurent series at the point identical with the given series except possibly for the coefficients of powers greater than 2g, where g is the genus of the surface.

Mathematical Subject Classification 2000
Primary: 30F99
Secondary: 30E10
Received: 1 July 1981
Revised: 28 April 1982
Published: 1 July 1983
Stephen Scheinberg