Idempotents et échantillonnage parcimonieux

Details Idempotents et échantillonnage parcimonieux Idempotents et échantillonnage parcimonieux

2011-09-14

arxiv.org/abs/1109.3179v1

Mathematics

Functional Analysis

\`a para\^itre aux Comptes rendus de l'Acad\'emie des sciences

Scientific paper

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.

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