Multiplier sequences and logarithmic mesh

Olga Katkova; Boris Shapiro; Anna Vishnyakova

doi:10.1016/j.crma.2010.11.031

Mathematical Analysis

Multiplier sequences and logarithmic mesh

Presented by: Jean-Pierre Kahane

Olga Katkova ¹; Boris Shapiro ²; Anna Vishnyakova ¹

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, Volume 349 (2011) no. 1-2, pp. 35-38.

Abstract
Résumé

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.

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.

Received: 2010-11-09
Accepted: 2010-11-30
Published online: 2010-12-22

DOI: 10.1016/j.crma.2010.11.031

Author's affiliations:

Olga Katkova ¹; Boris Shapiro ²; Anna Vishnyakova 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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     author = {Olga Katkova and Boris Shapiro and Anna Vishnyakova},
     title = {Multiplier sequences and logarithmic mesh},
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     pages = {35--38},
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     year = {2011},
     doi = {10.1016/j.crma.2010.11.031},
     language = {en},
}

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Olga Katkova; Boris Shapiro; Anna Vishnyakova. 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://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.11.031/

References
Cited by

[1] J. Borcea; P. Brändén Pólya–Schur master theorems for circular domains and their boundaries, Ann. of Math. (2), Volume 170 (2009) no. 1, pp. 465-492

[2] T. Craven; G. Csordas Problems and theorems in the theory of multiplier sequences, Serdica Math. J., Volume 22 (1996), pp. 515-524

[3] T. Craven; G. Csordas Multiplier sequences for fields, Illinois J. Math., Volume 21 (1977) no. 4, pp. 801-817

[4] N. Obreschkov Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963

[5] G. Pólya; J. Schur Über zwei Arten von Faktorenfolgen in der Theorie der algebraische Gleichungen, J. Reine Angew. Math., Volume 144 (1914), pp. 89-113

[6] A. Stoyanoff Sur un theoreme de M. Marcel Riesz, Nouv. Ann. Math., Volume 1 (1926), pp. 97-99

[7] G. Szegő Bemerkungen zu einem Satz von J.H. Grace über die Wurzeln algebraischer Gleichungen, Math. Z. (2), Volume 13 (1922), pp. 28-55

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