Comptes Rendus
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

Shunlong Luo1 ; Zhengmin Zhang2
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, Volume 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

Shunlong Luo 1 ; Zhengmin Zhang 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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Shunlong Luo; Zhengmin Zhang. An extremal problem for Fourier transforms of probabilities. Comptes Rendus. Mathématique, Volume 341 (2005) no. 5, pp. 293-296. doi : 10.1016/j.crma.2005.07.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.07.021/

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  • Z. Zhang Bounds for Characteristic Functions and Laplace Transforms of Probability Distributions, Theory of Probability Its Applications, Volume 56 (2012) no. 2, p. 350 | DOI:10.1137/s0040585x97985479
  • Ц Цзан; Z Zhang Bounds for characteristic functions and Laplace transforms of probability distributions, Теория вероятностей и ее применения, Volume 56 (2011) no. 2, p. 407 | DOI:10.4213/tvp4385
  • Xin He A problem on extreme magnitudes of characteristic functions: The general case, Statistics Probability Letters, Volume 78 (2008) no. 12, pp. 1514-1516 | DOI:10.1016/j.spl.2008.01.001 | Zbl:1155.60303
  • Zhengmin Zhang Inequalities for characteristic functions involving Fisher information, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 344 (2007) no. 5, pp. 327-330 | DOI:10.1016/j.crma.2007.01.008 | Zbl:1143.62001
  • Zhengmin Zhang An upper bound for characteristic functions of lattice distributions with applications to survival probabilities of quantum states, Journal of Physics A: Mathematical and Theoretical, Volume 40 (2007) no. 1, p. 131 | DOI:10.1088/1751-8113/40/1/007
  • Zhengmin Zhang A note on extreme magnitudes of characteristic functions, Statistics Probability Letters, Volume 77 (2007) no. 16, pp. 1641-1643 | DOI:10.1016/j.spl.2007.04.005 | Zbl:1132.60017

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