On the Mergelyan approximation property on pseudoconvex domains in ℂⁿ

Abstract

Abstract. Let Ω be a smoothly bounded pseudoconvex domain of finite type in C n . We prove the Mergelyan approximation property in various topologies on Ω when the estimates for ∂-equation are known in the corresponding topologies.

Holomorphic Approximation on Certain Weakly Pseudoconvex Domains in Cn

“…We finally estimate the L p -norm of f − f τ as τ → 0. Since f τ 0 = f , it follows from (12) and (11) that…”

“…Moreover, Cho proved in [11] that if D is a smoothly bounded pseudoconvex domain of finite type in C 2 , then every holomorphic function in the L p -Sobolev space W s,p (D), 1 < p < ∞, s ≥ 0, can be approximated on D by holomorphic functions on a neighborhood of D in the W s,p (D)-norm. In addition, he obtained the same result for the usual Lipschitz space.…”

Qualitative neural network approximation over R and C: Elementary proofs for analytic and polynomial activation

“…The situation in many complex variables is more complicated and not entirely understood. While multi-variate Mergelyan-type theorems [FGMN21,Gub15,Cho98] and related results like the Oka-Weil Theorem [Oka61, Wei35] have been obtained, there are obstructions to proving the statement in full generality. Notably, Diederich and Fornaess [DF76] constructed an example of a pseudoconvex domain D ⊆ C 2 with smooth boundary and a continuous function f : D → C such that f is holomorphic in D, but cannot be approximated uniformly by polynomials in D. Thus even for functions of multiple complex variables which can be shown to be holomorphic on a 'good' domain, there may be deep obstructions to approximation by both polynomials and holomorphic neural networks.…”