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.
