Universal functions with prescribed zeros and interpolation properties
The authors of this interesting paper study the existence of functions that are \textit{universal} with respect to differential operators and that are, at the same time, ``controlled'' by prescribed interpolation properties, including prescribed zeros and multiplicities. Let \(\Phi(z):=\sum_{n=0}^{\infty}a_nz^n\) denote an entire function of \textit{subexponential type} and consider the infinite-order differential operator \(\Phi(D)\), where \(D=d/dz\). The authors' aim is to prove the existence of holomorphic functions \(f\) on a simply connected domain \(\Omega\) that simultaneously satisfy the following conditions: (a) \(f\) is \(\Phi(D)\)-universal; (b) \(f\) has only zeros at the points of a given subset of \(\Omega\), with prescribed orders; and (c) \(f\) assumes prescribed values at prescribed points (cf. Section 3).
scientific article; zbMATH DE number 5665437