Random perturbations of exponential Riesz bases in $L^2(-\pi ,\pi )$

Chistyakov, Gennadii; Lyubarskii, Yura

Annales de l'Institut Fourier, Tome 47 (1997) no. 1, pp. 201-255.

Soient $\left\{{\lambda }_n\right\}\subset ℝ$ une suite donnée telle que le système exponentiel $\left\{\exp \right(i{\lambda }_{n}x\left)\right\}$ forme une base de Riesz dans $L^2(-\pi ,\pi )$ et $\left\{{\xi }_n\right\}$ une suite de variables aléatoires réelles indépendantes. On étudie les propriétés du système $\left\{\exp \right(i({\lambda }_n+{\xi }_n)x\left)\right\}$ 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 $\{{\lambda }_n+{\xi }_n\}$.

Let a sequence $\left\{{\lambda }_n\right\}\subset ℝ$ be given such that the exponential system $\left\{\exp \right(i{\lambda }_{n}x\left)\right\}$ forms a Riesz basis in $L^2(-\pi ,\pi )$ and $\left\{{\xi }_n\right\}$ be a sequence of independent real-valued random variables. We study the properties of the system $\{exp(i({\lambda }_n+{\xi }_n)x\left)\right\}$ 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 $\{{\lambda }_n+{\xi }_n\}$.

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