Holomorphic Approximation: The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan

Abstract

In this paper we survey the theory of holomorphic approximation, from the classical nineteenth century results of Runge and Weierstrass, continuing with the twentieth century work of Oka and Weil, Mergelyan, Vitushkin, and others, to the most recent ones on higher dimensional manifolds. The paper includes some new results and applications of this theory, especially to manifold-valued maps.