Doctor of Philosophy
Paul. M. Gauthier, Gordon J Sinnamon
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. Firstly, we show, however, that for every open Riemann surface R and every closed subset E of R; there is closed subset F of E, which approximates E extremely well, such that every function holomorphic on F can be approximated much better than uniformly by functions holomorphic on R. Secondly, given a function f 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. 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 reection principles of Schwarz and Caratheodory give conditions under which holomorphic functions extend holomorphically to the boundary and the theorem of Osgood-Caratheodory 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. Finally, in the last Chapter, we study similar questions on Riemann surfaces for holomorphic mappings.
Sharifi, Fatemeh, "Uniform Approximation on Riemann Surfaces" (2016). Electronic Thesis and Dissertation Repository. 3954.