On the ``universality'' of any holomorphic function
``Universal'' approximation properties for sequences of so called antiderivatives are studied in open sets \(O\subset {\mathbb{C}}\) with simply connected components. Let H(O) be the collection of all functions that are holomorphic on O and denote by \(I_ n(O,\phi)\) the family \[ \{\Phi | \quad \Phi \in H(O),\quad (d^ n/dz^ n)\Phi (z)=\phi (z)\text{ for all } z\in O\}. \] Any function \(\phi \in I_ n(O,\phi)\) is called an n-fold antiderivative of \(\phi\) on O. For such functions it is proved: (1) Let be given any open set \(U\subset O\) with simply connected components and any function \(f\in H(U)\). Then there exists a sequence \(\{\phi_ n^{(1)}\}\) with \(\phi_ n^{(1)}\in I_ n(O,\phi)\) that converges to f compactly on U. (2) Let be given any compact set \(B\subset O\) with connected complement and any function f that is continuous on B and holomorphic in its interior. Then there exists a sequence \(\{\phi_ n^{(2)}\}\) with \(\phi_ n^{(2)}\in I_ n(O,\phi)\) that converges to f uniformly on B. (3) Let be given any Lebesgue-measurable set \(E\subset O\) and any function f that is Lebesgue-measurable on E. Then there exists a sequence \(\{\phi_ n^{(3)}\}\) with \(\phi_ n^{(3)}\in I_ n(0,\phi)\) that converges to f almost everywhere on E. The proof uses results on approximation by ``strict sequences of antiderivatives'' [compare also the author in Constructive approximation 2, 179-187 (1986; Zbl 0615.30034)] and well known results on holomorphic functions like Runge's or Mergelyan's theorems. The author remarks that the topological assumptions on U, B, E in theorem 3 and the analytical assumptions made on the approximated functions cannot be weakened.
scientific article; zbMATH DE number 4017287
scientific article