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Comptes Rendus Mathématique
Volume 349, n° 19-20
pages 1073-1076 (novembre 2011)
Doi : 10.1016/j.crma.2011.08.015
Received : 18 August 2011 ;  accepted : 22 August 2011
Idempotents et échantillonnage parcimonieux
Idempotents and compressive sampling
 

Jean-Pierre Kahane
Laboratoire de mathématique, université Paris-sud, bâtiment 425, 91405 Orsay cedex, France 

Résumé

Comment reconstituer un signal, assimilé à une fonction x définie sur le groupe cyclique  , quʼon sait porté par T points, en nʼutilisant sa transformée de Fourier   que sur un ensemble Ω de fréquences ? Le procédé indiqué par Candès (2006) [[1]], Candès, Romberg et Tao (2006) [[2]] est lʼextrapolation minimale de   dans  . La note traite les questions suivantes : 1) Quand est–il vrai que ce procédé redonne tous les signaux portés par T points ? 2) Si lʼon choisit Ω par sélection aléatoire de points de  , N étant très grand, avec quelle probabilité obtient–on par ce procédé tous les signaux portés par T points ? 3) tous les signaux portés par un ensemble S donné ? 4) un signal donné ? Je donne des réponses à 1) et à 2) avec démonstrations, et à 3) sans démonstration. La réponse à 3) améliore les estimations de Candès, Romberg et Tao relatives à 4), la question quʼils traitent. Lʼidempotent K tel que   joue un rôle central.

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According to Candès (2006) [[1]], Candès, Romberg and Tao (2006) [[2]], a signal is represented as a function x defined on the cyclic group  . Assuming that it is carried by a set S consisting of T points, how to reconstruct x by using only a small set Ω of frequencies? The procedure of Candès, Romberg and Tao is the minimal extrapolation of   in  , when it exists. 1) When can we obtain in this way all signals carried by T points? 2) Choosing Ω by a random selection of points in   with N very large, give an estimate of the probability that the procedure works for all signals carried by T points 3) for all signals carried by a given set S 4) for a given signal. The answers to 1) and 2) are given with proofs and the answer to 3) without proof. Candès, Romberg and Tao answered question 4) and our answer to 3) improves their estimates. A key role is played by the idempotent K such that  .

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