On a family of extremal polynomials

Dmitrishin, Dmitriy; Smorodin, Andrey; Stokolos, Alex

doi:10.1016/j.crma.2019.06.010

Complex analysis

On a family of extremal polynomials
[Sur une famille de polynômes extrémaux]

Présenté par : the Editorial Board

Dmitrishin, Dmitriy ¹ ; Smorodin, Andrey ¹ ; Stokolos, Alex ²

¹ Odessa National Polytechnic University, 1 Shevchenko Ave., Odessa 65044, Ukraine
² Georgia Southern University, Statesboro, GA 30460, USA

Comptes Rendus. Mathématique, Tome 357 (2019) no. 7, pp. 591-596.

Résumé
Abstract

Pour une paire de polynômes trigonométriques $C (t) = \sum_{j = 1}^{N} a_{j} \cos j t$ , $S (t) = \sum_{j = 1}^{N} a_{j} \sin j t$ à coefficients réels avec la normalisation $a_{1} = 1$ , on trouve la valeur extrémale

\sup_{a_{2}, \dots, a_{N}} \min_{t} {C (t) : S (t) = 0} = - \frac{1}{4} \sec^{2} \frac{π}{N + 2} .

Une application en analyse géométrique complexe est montrée. On formule quelques conjectures pour les problèmes extrémaux sur les classes de polynômes.

For a pair of conjugate trigonometrical polynomials $C (t) = \sum_{j = 1}^{N} a_{j} \cos j t$ , $S (t) = {\sum_{j = 1}}^{N} a_{j} \sin j t$ with real coefficients and normalization $a_{1} = 1$ the following extremal value is found:

\sup_{a_{2}, \dots, a_{N}} \min_{t} {C (t) : S (t) = 0} = - \frac{1}{4} \sec^{2} \frac{π}{N + 2} .

An application of this result in geometric complex analysis is shown. Several conjectures for a number of extremal problems on classes of polynomials are suggested.

Reçu le : 2018-05-26
Accepté le : 2019-06-28
Publié le : 2019-07-01

DOI : 10.1016/j.crma.2019.06.010

Affiliations des auteurs :

Dmitrishin, Dmitriy ¹ ; Smorodin, Andrey ¹ ; Stokolos, Alex ²

¹ Odessa National Polytechnic University, 1 Shevchenko Ave., Odessa 65044, Ukraine
² Georgia Southern University, Statesboro, GA 30460, USA

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     title = {On a family of extremal polynomials},
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     pages = {591--596},
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Dmitrishin, Dmitriy; Smorodin, Andrey; Stokolos, Alex. On a family of extremal polynomials. Comptes Rendus. Mathématique, Tome 357 (2019) no. 7, pp. 591-596. doi : 10.1016/j.crma.2019.06.010. http://www.numdam.org/articles/10.1016/j.crma.2019.06.010/

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