2019/01/18 by Katkova, Olga, Tyaglov, Mikhail, Vishnyakova, Anna
#16C10 #26C10 #30C15 #30D15 #30D35 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1901.06398
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 Δθ, h(f)(z)=eiθf(z+ih)-e-iθf(z-ih), θ∈[0,π), h∈ℂ∖\0\, where f is a polynomial or an entire function of a certain kind, and prove that the roots of Δθ, h(f) are simple under some conditions. Moreover, we prove that the operator Δθ, 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 Δθ, h(p) as |h|→∞ is found for any complex polynomial p. Some other interesting roots preserving properties of the operator Δθ, h are also studied, and a few examples are presented.