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 19th century results of Runge and Weierstrass, continuing with the 20th century work of Oka and Weil, Mergelyan, Vituškin 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.

A function algebra providing new Mergelyan type theorems in several complex variables

“…Vitushkin [16] gave the best Mergelyan type theorem by providing necessary and sufficient conditions for the equality O(K) = A(K). However, in several complex variables the analogous theory is much less developed, see [4,11]. It is natural to investigate the case where K is a product of planar compact sets.…”

“…In particular, we have the celebrated theorems of Runge and Mergelyan. The present paper deals with approximation in several variables, where the situation is far from being understood [4,11].…”

Algebraic approximation and the Mittag-Leffler theorem for minimal surfaces

“…Indeed, just observe that convergence of the Weierstrass data is ensured in the proofs. Furthermore, in view of the recent result by Fornaess, Forstnerič, and Wold [14,Theorem 16] on Mergelyan approximation in the C r topology on admissible sets, it seems that the results in this paper can be extended by guaranteeing approximation of this class.…”

“…The aforementioned theorems admit several generalizations in complex analysis and algebraic geometry; we refer to the survey of Fornaess, Forstnerič, and Wold [14] for a review of this classical but still very active subject. Concerning meromorphic functions on compact Riemann surfaces, we recall the following extension of Runge's theorem, including interpolation, which dates back to the early decades of modern Riemann surface theory.…”

Hamiltonian Carleman approximation and the density property for coadjoint orbits

“…For information about Carleman approximation by functions, please see the recent survey [6]. Related to Theorem 1.1, it was proved in [14] that any diffeomorphism of R k can be approximated by holomorphic automorphisms of C n in the Carleman sense, provided that k < n, however, in that case R k was not left invariant.…”