In this Note we prove a new result about (finite) multiplier sequences, i.e. linear operators acting diagonally in the standard monomial basis of and sending polynomials with all real roots to polynomials with all real roots. Namely, we show that any such operator does not decrease the logarithmic mesh when acting on an arbitrary polynomial having all roots real and of the same sign. The logarithmic mesh of such a polynomial is defined as the minimal quotient of its consecutive roots taken in the non-decreasing order of their absolute values.
Les multiplicateurs considérés dans cette Note sont les opérateurs linéaires qui agissent diagonalement sur muni de sa base standard (les monômes) et qui transforment les polynômes à racines réelles en polynômes à racines réelles. Nous montrons qu'un tel opérateur, appliqué à un polynôme dont toutes les racines sont réelles et de même signe, ne diminue pas la maille logarithmique, c'est-à-dire le minimum du quotient de deux racines consécutives dans l'ordre croissant des valeurs absolues.
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@article{CRMATH_2011__349_1-2_35_0,
author = {Katkova, Olga and Shapiro, Boris and Vishnyakova, Anna},
title = {Multiplier sequences and logarithmic mesh},
journal = {Comptes Rendus. Math\'ematique},
pages = {35--38},
publisher = {Elsevier},
volume = {349},
number = {1-2},
year = {2011},
doi = {10.1016/j.crma.2010.11.031},
language = {en},
url = {https://www.numdam.org/articles/10.1016/j.crma.2010.11.031/}
}
TY - JOUR AU - Katkova, Olga AU - Shapiro, Boris AU - Vishnyakova, Anna TI - Multiplier sequences and logarithmic mesh JO - Comptes Rendus. Mathématique PY - 2011 SP - 35 EP - 38 VL - 349 IS - 1-2 PB - Elsevier UR - https://www.numdam.org/articles/10.1016/j.crma.2010.11.031/ DO - 10.1016/j.crma.2010.11.031 LA - en ID - CRMATH_2011__349_1-2_35_0 ER -
%0 Journal Article %A Katkova, Olga %A Shapiro, Boris %A Vishnyakova, Anna %T Multiplier sequences and logarithmic mesh %J Comptes Rendus. Mathématique %D 2011 %P 35-38 %V 349 %N 1-2 %I Elsevier %U https://www.numdam.org/articles/10.1016/j.crma.2010.11.031/ %R 10.1016/j.crma.2010.11.031 %G en %F CRMATH_2011__349_1-2_35_0
Katkova, Olga; Shapiro, Boris; Vishnyakova, Anna. Multiplier sequences and logarithmic mesh. Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 35-38. doi : 10.1016/j.crma.2010.11.031. https://www.numdam.org/articles/10.1016/j.crma.2010.11.031/
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