Hermite-Poulain theorems for linear finite difference operators

Abstract

We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form $$ Δ_{θ, h}(f)(z)=e^{iθ}f(z+ih)-e^{-iθ}f(z-ih), \quadθ\in[0,π),\ \ h\in\mathbb{C}\setminus\{0\}, $$ where $f$ is a polynomial or an entire function of a certain kind, and prove that the roots of $Δ_{θ, h}(f)$ are simple under some conditions. Moreover, we prove that the operator $Δ_{θ, h}$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of $Δ_{θ, h}(p)$ as $|h|\to\infty$ is found for any complex polynomial $p$. Some other interesting roots preserving properties of the operator $Δ_{θ, h}$ are also studied, and a few examples are presented.