#### Degree

Doctor of Philosophy

#### Program

Mathematics

#### Supervisor

Paul. M. Gauthier

#### 2nd Supervisor

Gordon J Sinnamon

Joint Supervisor

#### Abstract

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.

#### Recommended Citation

Sharifi, Fatemeh, "Uniform Approximation on Riemann Surfaces" (2016). *Electronic Thesis and Dissertation Repository*. 3954.

https://ir.lib.uwo.ca/etd/3954