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Elements of Pólya-Schur theory in the finite difference setting

2016
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Proceedings of the American Mathematical Society
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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

doi:10.1090/proc/13115
fatcat:qap2etdlejenrfnfojfecf27ae