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Elements of Pólya-Schur theory in the finite difference setting

Authors: Petter Brändén, Ilia Krasikov and Boris Shapiro
Journal: Proc. Amer. Math. Soc. 144 (2016), 4831-4843
MSC (2010): Primary 26C10; Secondary 30C15
Published electronically: April 19, 2016
MathSciNet review: 3544533
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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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Petter Brändén
Affiliation: Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden

Ilia Krasikov
Affiliation: Department of Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, United Kingdom

Boris Shapiro
Affiliation: Department of Mathematics, Stockholm University, S-10691, Stockholm, Sweden

Keywords: Finite difference operators, hyperbolicity preservers, mesh
Received by editor(s): October 21, 2014
Received by editor(s) in revised form: September 30, 2015, and January 15, 2016
Published electronically: April 19, 2016
Communicated by: Walter Van Assche
Article copyright: © Copyright 2016 American Mathematical Society

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