Variantes sur un théorème de Candès, Romberg et Tao
Résumé
Here is a general question in harmonic analysis. Given two locally com- pact abelian groups in duality, G and ˆG, L1 (G) is mapped by the Fourier transformation onto A( ˆG), the Wiener algebra of ˆG. We want to recons- truct an element x of L1 (G) from the restriction of its Fourier transform ˆx to some subset E of ˆG. That is possible when ˆx is the minimal extension in A( ˆG) of its restriction to E. When is that realized ? The subject originates from signal's theory and the basic reference is the theory developed by Candès, Romberg and Tao (2006), where s denotes a signal, G and ˆG, both represented by ZN , are respectively the spaces of times and of frequencies. Here E is a set of frequencies ω and it is denoted by Ω. The condition on the signal x is that it is carried by a time-set S consisting of T points. In order to have Ω small compared to N , the main idea is to choose it randomly. Then a convenient relation between T , N , and the size of ω insures that the condition on ˆx is satisfied with a high probability. In the present article we want to reconstruct not only one signal x but all signals carried by the same set S , or even all signals carried by an unspecified set of T points. Statement V 1 is a deterministic answer to the last question, in the form of a simple condition on the idempotent K whose spectrum is Ω. Statement V 1, V 1 and V " "1 use randomness and provide explicit probabilistic conditions. In the same way statements V 2, V 2 and V 2 give explicit probabilistic conditions for the reconstruction of all signals carried by a set S of T points ; they extend and improve the theorem of Candès, Romberg and Tao. Statement V 4 has nothing to do with signal's theory and it is given without proof. It applies the general idea to the reconstruction of a lacunary trigonometric series when its sum is given on a small interval.
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