On uniform approximation by \(n\)-analytic functions on closed sets in \(\mathbb C\). (English. Russian original) Zbl 1069.30076
Izv. Math. 68, No. 3, 447-459 (2004); translation from Izv. Ross. Akad. Nauk Ser. Mat. 68, No. 3, 15-28 (2004).
Let \(n\) be a positive integer. A function \(f\) is called \(n\)-analytic on an open set \(G\) in the complex plane if \(f\in C^n(G)\) and \(\bar{\partial}^nf=0\) in \(G\). Such a function has a unique representation
\[
f(z)=f_0(z)+\bar{z}f_1(z)+\dots +\bar{z}^{n-1}f_{n-1}(z),
\]
where \(f_0,f_1,\dots f_{n-1}\) are holomorphic functions on \(G\). The authors provide necessary and/or sufficient conditions on a closed set \(F\) for any function \(f\) continuous on \(F\) and \(n\)-analytic in the interior of \(F\), to be the uniform limit on \(F\) of a sequence of \(n\)-analytic entire or meromorphic functions. The proofs involve the notions of Carathéodory and Nevanlinna domains, and the classical approximation theorems (Mergelyan, Arakelyan etc).
Reviewer: Dimitrios Betsakos (Thessaloniki)
MSC:
30G30 | Other generalizations of analytic functions (including abstract-valued functions) |
30E10 | Approximation in the complex plane |