Mathematical Analysis
Multiplier sequences and logarithmic mesh
[Multiplicateurs et mailles logarithmiques]

Présenté par : Jean-Pierre Kahane

Katkova, Olga1 ; Shapiro, Boris2 ; Vishnyakova, Anna1
1 Department of Mechanics & Mathematics, Kharkov National University, 4 Svobody Sq., Kharkov, 61077, Ukraine
2 Department of Mathematics, Stockholm University, 10691, Stockholm, Sweden
Comptes Rendus. Mathématique, Tome 349 (2011) no. 1-2, pp. 35-38.

Les multiplicateurs considérés dans cette Note sont les opérateurs linéaires qui agissent diagonalement sur R[x] 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.

In this Note we prove a new result about (finite) multiplier sequences, i.e. linear operators acting diagonally in the standard monomial basis of R[x] 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.11.031
Katkova, Olga 1 ; Shapiro, Boris 2 ; Vishnyakova, Anna 1

1 Department of Mechanics & Mathematics, Kharkov National University, 4 Svobody Sq., Kharkov, 61077, Ukraine
2 Department of Mathematics, Stockholm University, 10691, Stockholm, Sweden 
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Katkova, Olga; Shapiro, Boris; Vishnyakova, Anna. Multiplier sequences and logarithmic mesh. Comptes Rendus. Mathématique, Tome 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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