# Weakly Increasing Zero-Diminishing Sequences

Bakan, Andrew; Craven, Thomas; Csordas, George; Golub, Anatoly

Serdica Mathematical Journal (1996)

- Volume: 22, Issue: 4, page 547-570
- ISSN: 1310-6600

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topBakan, Andrew, et al. "Weakly Increasing Zero-Diminishing Sequences." Serdica Mathematical Journal 22.4 (1996): 547-570. <http://eudml.org/doc/11651>.

@article{Bakan1996,

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.},

author = {Bakan, Andrew, Craven, Thomas, Csordas, George, Golub, Anatoly},

journal = {Serdica Mathematical Journal},

keywords = {Weakly Increasing Sequences; Zero-Diminishing Sequences; Zeros of Entire Functions; Interpolation; weakly increasing sequences; zero-diminishing sequences; zeros of entire functions; interpolation},

language = {eng},

number = {4},

pages = {547-570},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {Weakly Increasing Zero-Diminishing Sequences},

url = {http://eudml.org/doc/11651},

volume = {22},

year = {1996},

}

TY - JOUR

AU - Bakan, Andrew

AU - Craven, Thomas

AU - Csordas, George

AU - Golub, Anatoly

TI - Weakly Increasing Zero-Diminishing Sequences

JO - Serdica Mathematical Journal

PY - 1996

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 22

IS - 4

SP - 547

EP - 570

AB - 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.

LA - eng

KW - Weakly Increasing Sequences; Zero-Diminishing Sequences; Zeros of Entire Functions; Interpolation; weakly increasing sequences; zero-diminishing sequences; zeros of entire functions; interpolation

UR - http://eudml.org/doc/11651

ER -

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