Classification theorems for operators preserving zeros in a strip

Abstract

We characterize all linear operators which preserve spaces of entire functions whose zeros lie in a closed strip. Necessary and sufficient conditions are obtained for the related problem with real entire functions, and some classical theorems of de Bruijn and P\'olya are extended. Specifically, we reveal new differential operators which map real entire functions whose zeros lie in a strip into real entire functions whose zeros lie in a narrower strip; this is one of the properties that characterize a "strong universal factor" as defined by de Bruijn. Using elementary methods, we prove a theorem of de Bruijn and extend a theorem of de Bruijn and Ilieff which states a sufficient condition for a function to have a Fourier transform with only real zeros.