Elements of Polya-Schur theory in finite difference setting

In this note we attempt to develop an analog of P\'olya-Schur theory describing the class of univariate hyperbolicity preservers 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, finite difference versions of the classical Hermite-Poulain theorem and generalized Laguerre inequalities are obtained.