Weakly Increasing Sequences Zero-Diminishing Sequences Zeros of Entire Functions Interpolation

Issue Date:

1996

Publisher:

Institute of Mathematics and Informatics Bulgarian Academy of Sciences

Citation:

Serdica Mathematical Journal, Vol. 22, No 4, (1996), 547p-570p

Abstract:

The following problem, suggested by Laguerre’s Theorem (1884),
remains open: Characterize all real sequences {μk} k=0...∞
which have the zero-diminishing property; that is, if k=0...n, p(x) = ∑(ak x^k) is any P real polynomial, then
k=0...n, p(x) = ∑(μk ak x^k) has no more real zeros than p(x).
In this paper this problem is solved under the additional assumption of a weak
growth condition on the sequence {μk} k=0...∞, namely lim n→∞ | μn |^(1/n) < ∞.
More precisely, it is established that the real sequence {μk} k≥0 is a weakly increasing zerodiminishing
sequence if and only if there exists σ ∈ {+1,−1} and an entire function
n≥1, Φ(z)= be^(az) ∏(1+ x/αn), a, b ∈ R^1, b =0, αn > 0 ∀n ≥ 1, ∑(1/αn) < ∞, such that µk = (σ^k)/Φ(k), ∀k ≥ 0.