Approximation by antiderivatives
Suppose that \(\Phi\) is an entire function and define a sequence \(\Phi^{(-n)}\) of so called antiderivatives of \(\Phi\) by: \[ \Phi^{(0)}(z)=\Phi (z),\quad (d/dz)\Phi^{(-n-1)}(z)=\Phi^{(-n)}(z) \] for all \(n\in N_ 0\). For such a sequence \textit{Ch. Blair} and \textit{L. Rubel} proved in Enseign. Math., II. Ser. 30, 269-274 (1984; Zbl 0559.30033). For any entire f, there exists a subsequence \(\{n_ k\}\) of the natural numbers such that \(\{\Phi^{(-n_ k)}\}\) converges to f compactly on \({\mathbb{C}}\). In this paper, it is shown, that not only an entire function \(\Phi\), but also any function \(\Phi\) that is holomorphic on an arbitrary open set \(\Omega\subset {\mathbb{C}}\) having simply connected components, has such a ''universal'' approximation property. Also the approximation properties of subsequences \(\{\Phi^{(-n_ k)}\}\) on compact, open, or measurable subsets of \(\Omega\) are discussed.
scientific article; zbMATH DE number 3996077
scientific article