Idempotents et \'echantillonnage parcimonieux

Abstract

Following Cand\`es, 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 Cand\`es, Romberg and Tao in order to get new results. The idempotent whose Fourier transform is the indicator function of W plays a central role.