Methods of harmonic analysis in nonlinear dynamics

Dmitrishin, Dmitriy; Khamitova, Anna

doi:10.1016/j.crma.2013.05.009

Harmonic Analysis/Dynamical Systems

Methods of harmonic analysis in nonlinear dynamics
[Méthodes dʼanalyse harmonique en dynamique non linéaire]

Présenté par : Jean-Pierre Kahane

Dmitrishin, Dmitriy ¹ ; Khamitova, Anna ²

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

Comptes Rendus. Mathématique, Tome 351 (2013) no. 9-10, pp. 367-370.

Résumé
Abstract

Pour un couple de polynômes trigonométriques $C (t) = \sum_{j = 1}^{N} a_{j} \cos j t$ , $S (t) = \sum_{j = 1}^{N} a_{j} \sin j t$ , normalisés par la condition $\sum_{j = 1}^{N} a_{j} = 1$ , on a la formule extrémale suivante :

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

On donne une application de ce résultat en théorie du contrôle à des systèmes non linéaires discrets.

For a pair of conjugate trigonometric polynomials $C (t) = \sum_{j = 1}^{N} a_{j} \cos j t$ , $S (t) = \sum_{j = 1}^{N} a_{j} \sin j t$ , normalized by the condition $\sum_{j = 1}^{N} a_{j} = 1$ , the following extremal value is found:

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

An application of this result in the control theory for nonlinear discrete systems is shown.

Reçu le : 2013-04-06
Accepté le : 2013-05-23
Publié le : 2013-05-01

DOI : 10.1016/j.crma.2013.05.009

Affiliations des auteurs :

Dmitrishin, Dmitriy ¹ ; Khamitova, Anna 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 = {Methods of harmonic analysis in nonlinear dynamics},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {367--370},
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Dmitrishin, Dmitriy; Khamitova, Anna. Methods of harmonic analysis in nonlinear dynamics. Comptes Rendus. Mathématique, Tome 351 (2013) no. 9-10, pp. 367-370. doi : 10.1016/j.crma.2013.05.009. http://www.numdam.org/articles/10.1016/j.crma.2013.05.009/

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