Random perturbations of exponential Riesz bases in ${L}^{2}\left(-\pi ,\pi \right)$
Annales de l'Institut Fourier, Volume 47 (1997) no. 1, p. 201-255

Let a sequence $\left\{{\lambda }_{n}\right\}\subset ℝ$ be given such that the exponential system $\left\{\mathrm{exp}\phantom{\rule{0.166667em}{0ex}}\left(i{\lambda }_{n}x\right)\right\}$ forms a Riesz basis in ${L}^{2}\left(-\pi ,\pi \right)$ and $\left\{{\xi }_{n}\right\}$ be a sequence of independent real-valued random variables. We study the properties of the system $\left\{exp\left(i\left({\lambda }_{n}+{\xi }_{n}\right)x\right)\right\}$ as well as related problems on estimation of entire functions with random zeroes and also problems on reconstruction of bandlimited signals with bandwidth $2\pi$ via their samples at the random points $\left\{{\lambda }_{n}+{\xi }_{n}\right\}$.

Soient $\left\{{\lambda }_{n}\right\}\subset ℝ$ une suite donnée telle que le système exponentiel $\left\{\mathrm{exp}\phantom{\rule{0.166667em}{0ex}}\left(i{\lambda }_{n}x\right)\right\}$ forme une base de Riesz dans ${L}^{2}\left(-\pi ,\pi \right)$ et $\left\{{\xi }_{n}\right\}$ une suite de variables aléatoires réelles indépendantes. On étudie les propriétés du système $\left\{\mathrm{exp}\phantom{\rule{0.166667em}{0ex}}\left(i\left({\lambda }_{n}+{\xi }_{n}\right)x\right)\right\}$ ainsi que des problèmes reliés aux estimations des fonctions entières ayant des zéros aléatoires, et des problèmes de reconstitution de signaux avec un spectre de largeur $2\pi$ à l’aide de valeurs de ces signaux dans des points aléatoires $\left\{{\lambda }_{n}+{\xi }_{n}\right\}$.

@article{AIF_1997__47_1_201_0,
author = {Chistyakov, Gennadii and Lyubarskii, Yura},
title = {Random perturbations of exponential Riesz bases in $L^2(-\pi ,\pi )$},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {47},
number = {1},
year = {1997},
pages = {201-255},
doi = {10.5802/aif.1565},
zbl = {0860.42023},
mrnumber = {98c:42028},
language = {en},
url = {http://www.numdam.org/item/AIF_1997__47_1_201_0}
}

Chistyakov, Gennadii; Lyubarskii, Yura. Random perturbations of exponential Riesz bases in $L^2(-\pi ,\pi )$. Annales de l'Institut Fourier, Volume 47 (1997) no. 1, pp. 201-255. doi : 10.5802/aif.1565. http://www.numdam.org/item/AIF_1997__47_1_201_0/

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