Elements of PÓlya-schur theory in the finite difference setting

Petter Brändén, Ilia Krasikov, Boris Shapiro

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)4831-4843
Number of pages13
JournalProceedings of the American Mathematical Society
Issue number11
StatePublished - 2016
Externally publishedYes


  • Finite difference operators
  • Hyperbolicity preservers
  • Mesh


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