Approximation by Solutions of Elliptic Equations and Extension of Subharmonic Functions

Holomorphic approximation: the legacy of Weierstrass, Runge, Oka-Weil, and Mergelyan

“…In conclusion, we mention that many of the results on holomorphic approximation, presented in this and the previous two sections, have been generalized to solutions of more general elliptic differential equations in various Banach space norms; see in particular J. Verdera [158], P. Paramonov and J. Verdera [131], A. Boivin, P. Gauthier and P. Paramonov [25], and P. Gauthier and P. Paramonov [72].…”

“…In this and the following two sections we survey the main achievements of the classical holomorphic approximation theory. More comprehensive surveys of this subject are available in [24,67,68,69,70,72,74,76,174], among other sources.…”

“…There exist a number of surveys on holomorphic approximation theory; see e.g. [24,67,68,69,70,72,74,76,77,105,174], among others. However, ours seems the first attempt at a unified picture, from the highlights of the classical theory to results in several variables and for manifold-valued maps.…”

Approximation of continuous functions on a compact set by solutions of elliptic equations. Quantitative results

“…Up to now, there are numerous studies of approximation by polynomials, analytic, and harmonic functions, with common feature of these functions being that they possess a lot of algebraic and analytic structures, which enrich possible methods of analysis (Math.Sci.Net contains several thousand references on this topic.) A detailed discussion, with a number of references, can be found in the books [2], [11], [8], [23]; some most recent developments are presented, in particular, in papers by P. Paramonov, P. Gauthier and their co-operators (see, e.g., their latest publications [20], [12] and references therein).…”