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

$$\begin{aligned} \Delta _{\theta , h}(f)(z)=e^{i\theta }f(z+ih)-e^{-i\theta }f(z-ih), \quad \theta \in [0,\pi ),\ \ h\in \mathbb {C}{\setminus }\{0\}, \end{aligned}$$

where f is a polynomial or an entire function of a certain kind, and prove that the roots of $\Delta _{\theta , h}(f)$ are simple under some conditions. Moreover, we prove that the operator $\Delta _{\theta , h}$ 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 $\Delta _{\theta , h}(p)$ as $|h|\rightarrow \infty$ are found for any complex polynomial p. Some other interesting roots preserving properties of the operator $\Delta _{\theta , h}$ are also studied, and a few examples are presented.