  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.
