Uniform approximation by entire functions that are all bounded on a given set
The famous approximation theorem of Arakelyan (cf. \textit{D. Gaier}: Lectures on complex approximation [1980, Zbl 0442.30038)]) gives a complete (topological) description of those closed sets \(F\) in the complex plane which admit uniform approximation of functions \(f\in A(F)\) (i.e. continuous on \(F\) and holomorphic in the interior of \(F)\) by entire functions. The authors study how far this approximation can be done by entire functions which are bounded on another given closed set \(G\subset\mathbb{C}\). They prove that this can be done iff \(G\) is contained in some closed set \(G_1\neq\mathbb{C}\) with \(G\subset G_1\neq \mathbb{C}\) where \(F\) and \(G_1\) both fulfill the condition of Arakelyan's theorem and \(F\cap G_1\) is bounded.
scientific article; zbMATH DE number 1268074