Linear finite difference operators preserving Laguerre-P\'olya class

TL;DR

This paper characterizes all finite difference operators that preserve the Laguerre-Pólya class of entire functions, providing a complete description of such operators in terms of complex functions and shifts.

Contribution

It offers a full classification of finite difference operators of a specific form that maintain the Laguerre-Pólya class, advancing understanding in complex analysis.

Findings

Complete description of difference operators preserving Laguerre-Pólya class

Identification of conditions on functions M_1, M_2, and h

Extension of classical results in entire function theory

Abstract

We completely describe all finite difference operators of the form $$ \Delta_{M_1, M_2, h}(f)(z)=M_1(z) f(z+h) + M_2(z) f(z-h) $$ preserving the Laguerre-P\'olya class of entire functions. Here $M_1$ and $M_2$ are some complex functions and $h$ is a nonzero complex number.