The Oka–Weil theorem states that if X is a Stein space and K is a
compact-convex subset of X, then every holomorphic function in an
open neighborhood of K can be approximated uniformly on K by holomorphic functions on (i.e. by polynomials).[1]
Applications
Since
Runge's theorem may not hold for several complex variables, the Oka–Weil theorem is often used as an approximation theorem for several complex variables. The
Behnke–Stein theorem was originally proved using the Oka–Weil theorem.
Agler, Jim; McCarthy, John E. (2015). "Global Holomorphic Functions in Several Noncommuting Variables". Canadian Journal of Mathematics. 67 (2): 241–285.
arXiv:1305.1636.
doi:
10.4153/CJM-2014-024-1.
S2CID120834161.