Uniform Approximation on Riemann Surfaces

This thesis consists of three contributions to the theory of complex approximation on Riemann surfaces. It is known that if E is a closed subset of an open Riemann surface R and f is a holomorphic function on a neighbourhood of E, then it is “usually” not possible to approximate f uniformly by functions holomorphic on all of R. In Chapter 2, we show, however, that for every open Riemann surface R and every closed subset E ⊂ R, there is a closed subset F ⊂ E, which approximates E extremely well, and has the following property. Every function holomorphic on F can be approximated tangentially (much better than uniformly) by functions holomorphic on R. In Chapter 3, given a function f : E → C from a closed subset of a Riemann surface R to the Riemann sphere C, we seek to approximate f in the spherical distance by functions meromorphic on R. As a consequence we generalize a recent extension of Mergelyan’s theorem, due to Fragoulopoulou, Nestoridis and Papadoperakis [3.13]. The problem of approximating by meromorphic functions pole-free on E is equivalent to that of approximating by meromorphic functions zero-free on E, which in turn is related to Voronin’s spectacular universality theorem for the Riemann zeta-function. The reflection principles of Schwarz and Carathéodory give conditions under which holomorphic functions extend holomorphically to the boundary and the theorem of Osgood-Carathéodory states that a one-to-one conformal mapping from the unit disc to a Jordan domain extends to a homeomorphism of the closed disc onto the closed Jordan domain. In Chapter 4, we study similar questions on Riemann surfaces for holomorphic mappings.