no. 5
Harmonic Analysis
An extremal problem for Fourier transforms of probabilities
[Un problème extrêmal pour transformée de Fourier de probabilité]

Présenté par : Paul Malliavin

Luo, Shunlong1 ; Zhang, Zhengmin2
1 Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, PR China
2 School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada
Comptes Rendus. Mathématique, Tome 341 (2005) no. 5, pp. 293-296.

On résout un problème extrêmal concernant des fonctions caractéristiques soumises à la condition de s'annuler en un point fixé. L'origine du problème est l'étude de l'amplitude de survie d'un état quantique dans la dynamique de Schrödinger, et la solution exprime un phénomène curieux dans l'évolution des systèmes quantiques.

We study an extremal problem concerning the supremum of the Fourier transforms (characteristic functions) of probability distributions under the constraint that the Fourier transforms vanish at a fixed point. This problem arises from the investigation of the survival amplitudes of quantum states driven by Schrödinger dynamics, and has general and curious implications for the evolution pictures of quantum systems.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.07.021
Luo, Shunlong 1 ; Zhang, Zhengmin 2

1 Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, PR China
2 School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada 
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Luo, Shunlong; Zhang, Zhengmin. An extremal problem for Fourier transforms of probabilities. Comptes Rendus. Mathématique, Tome 341 (2005) no. 5, pp. 293-296. doi : 10.1016/j.crma.2005.07.021. http://www.numdam.org/articles/10.1016/j.crma.2005.07.021/

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[2] Fryntov, A.E. An extremal problem in the theory of Hermitian positive functions, Funkcional. Anal. i Priložen., Volume 10 (1976), pp. 91-92 (in Russian). English translation Functional Anal. Appl., 10, 1976, pp. 80-82

[3] Holland, G.H.; Sitaram, A. The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., Volume 3 (1997), pp. 207-236

[4] Lukacs, E. Characteristic Functions, Griffin, London, 1970

[5] Luo, S.L.; Zhang, Z.M. Estimating the first zero of a characteristic function, C. R. Acad. Sci. Paris, Ser. I, Volume 338 (2004), pp. 203-206

[6] Peres, A. Quantum Theory: Concepts and Methods, Kluwer, Dordrecht, 1993

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