Doctor of Philosophy
This thesis is of two parts: At the first part (Chapters 1 and 2) we study some spaces of holomorphic k-differentials on open Riemann surfaces, and obtain some observations about these spaces, then we obtain two main theorems about the kernel of Poincar\'e series map. In the second part (Chapters 3 and 4), we study holomorphic approximation on closed subsets of non-compact Riemann surfaces. We add a condition to the Extension Theorem and fixing its proof. Extension Theorem was first stated and proved by G. Schmieder, but there are few examples, where the theorem fails. That is slightly effecting a class of closed sets (which is called weakly of infinite genus).
Askaripour, Nadya, "Holomorphic k-differentials and holomorphic approximation on open Riemann surfaces" (2010). Electronic Thesis and Dissertation Repository. 9.