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 $${\Delta }_{\theta ,h}(f)(z)=e^{i\theta }f(z+ih)-e^{-i\theta }f(z-ih),\theta \in [0,\pi ),h\in \mathbb{C}\setminus \{0\},$$ where $f$ is a polynomial or an entire function of a certain kind, and prove that the roots of ${\Delta }_{\theta ,h}(f)$ 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 $|h|\to \infty$ is 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.

27 pages. This is an extended version of a preprint by the same authors

References in corpus (5)

• Polynomials, roots, and interlacing
• Complex zero strip decreasing operators
• Self-interlacing polynomials
• Linear finite difference operators preserving Laguerre-Pólya class
• Note on the location of zeros of polynomials