Arakelian's theorem

Abstract

[en] The approximation by rational functions and polynomials is one of the topics that has been studied for a long time. The aim of this text is to study the uniform approximation by rational functions and polynomials based on three theorems: Runge, Mergelyan, and Arakelian. The first one concerns uniform approximation by rational functions on compact sets. Mergelyan’s theorem is a generalization of Runge’s theorem. Finally, Arakelian’s theorem deals with uniform approximation by entire functions on possibly unbounded closed sets. We provide the proofs of these theorems and furthermore, we state connexions between them.