On a family of extremal polynomials

Dmitriy Dmitrishin; Andrey Smorodin; Alex Stokolos

doi:10.1016/j.crma.2019.06.010

Complex analysis

On a family of extremal polynomials

Presented by: the Editorial Board

Dmitriy Dmitrishin ¹; Andrey Smorodin ¹; Alex Stokolos ²

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

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

Abstract
Résumé

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.

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.

Received: 2018-05-26
Accepted: 2019-06-28
Published online: 2019-07-01

DOI: 10.1016/j.crma.2019.06.010

Author's affiliations:

Dmitriy Dmitrishin ¹; Andrey Smorodin ¹; Alex Stokolos 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},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {591--596},
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     year = {2019},
     doi = {10.1016/j.crma.2019.06.010},
     language = {en},
}

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Dmitriy Dmitrishin; Andrey Smorodin; Alex Stokolos. On a family of extremal polynomials. Comptes Rendus. Mathématique, Volume 357 (2019) no. 7, pp. 591-596. doi : 10.1016/j.crma.2019.06.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.06.010/

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