Elements of Pólya-Schur theory in the finite difference setting

Brändén, Petter; Krasikov, Ilia; Shapiro, Boris

doi:10.1090/proc/13115

Elements of Pólya-Schur theory in the finite difference setting
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by Petter Brändén, Ilia Krasikov and Boris Shapiro PDF

Proc. Amer. Math. Soc. 144 (2016), 4831-4843 Request permission

Abstract:

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 difference version of the classical Hermite-Poulain theorem and several results about discrete multiplier sequences.

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