Holomorphic and harmonic approximation on Riemann surfaces (Q2758405)

This article is a survey of the last hundred years in holomorphic and harmonic approximation. The first part on holomorphic approximation starts with Runge's theorem and presents different approaches to this basic result. Mergelyan's theorem as an essential extension of this classical result with applications and Vitushkin's criterion of rational approximation on compact plane sets follow and the generalizations on closed plane sets, due to A. Roth, Arakelian and Nersesian are shortly mentioned.NEWLINENEWLINENEWLINEThe main purpose is approximation on Riemann surfaces, which begun with the theorem of Behnke and Stein as the analogy of Runge's theorem on non-compact Riemann surfaces. Here Bishop's theorem is the generalization of Mergelyan's result and the transfer to Riemann surfaces of Arakelian's approximation theorem is due to Scheinberg (following Arakelian's idea) and Gauthier and Hengartner (following Roth's approach). In this results the approximation on a closed subset \(E\), of the Riemann surface \(R\) requires that \(E\) is essentially of finite genus, which means that \(E\) can be covered by pairwise disjoint open sets of finite genus. A sufficient condition without this substantial restriction given by the reviewer is finally mentioned. Some instructive examples are added. The theory of harmonic approximations (on Riemann surfaces) is closely linked with holomorphic approximation. The known results are relatively detailed described in the second part.NEWLINENEWLINENEWLINEThis article is recommended as a base of seminars on the beautiful field of complex approximation theory.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00045].