Universal Taylor series in simply connected domains
Let \(\Omega\) be a simply connected domain in the complex plane. The authors prove the existence of functions \(f\) holomorphic in \(\Omega\) which extend smoothly to some prescribed part \(S\) of the boundary \(\partial\Omega\) and having the property that, for every \(\zeta\in\Omega\), the sequence of partial sums \(S_n(f,\zeta)\) of the Taylor expansions with center \(\zeta\) exhibits universal overconvergence outside of \(\Omega\cup S\). This complements earlier results dealing with the special cases \(S = \emptyset\), that is universal overconvergence takes place outside of \(\Omega\), and \(S = \partial\Omega\), that is, \(f\) extends smoothly to \(\overline{\Omega}\).
scientific article; zbMATH DE number 5152625
scientific article