Idempotents et échantillonnage parcimonieux

Following Candès, Romberg and Tao (IEEE Transactions on Information Theory 20,2 (2006) 489-509) a signal is represented as a function x defined on the cyclic group G = Z / NZ, and assuming that it is carried by a set S consisting of T points, we want to reconstruct x using only a small set W of frequencies. The procedure is the minimal extrapolation of the restriction of the Fourier transform of x to W in the Wiener algebra of the dual of G , and the condition for that to work is a relation between x and W. The note gives conditions on S and W, and conditions on T and W for it to work, and uses random choices in another way than Candès, Romberg and Tao in order to get new results. The idempotent whose Fourier transform is the indicator function of W plays a central role.

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