Elements of Pólya-Schur theory in the finite difference setting

Petter Brändén, Ilia Krasikov, Boris Shapiro
2016 Proceedings of the American Mathematical Society  
The Pólya-Schur theory describes the class of hyperbolicity preservers, i.e., the class of linear operators acting on univariate polynomials and preserving real-rootedness. We attempt to develop an analog of Pólya-Schur theory in the setting of linear finite difference operators. We study the class of linear finite difference operators preserving the set of real-rooted polynomials whose mesh (i.e., the minimal distance between the roots) is at least one. In particular, we prove a finite
more » ... ve a finite difference version of the classical Hermite-Poulain theorem and several results about discrete multiplier sequences.
doi:10.1090/proc/13115 fatcat:qap2etdlejenrfnfojfecf27ae