Idempotents et \'echantillonnage parcimonieux

TL;DR

This paper explores conditions for signal reconstruction on cyclic groups using sparse frequency sampling, emphasizing the role of idempotents and randomization to improve upon previous methods.

Contribution

It introduces new conditions on sampling sets and frequency subsets, utilizing idempotents and alternative randomization techniques for signal reconstruction.

Findings

Derived new conditions for successful reconstruction

Utilized idempotents to analyze sampling and frequency sets

Applied novel randomization methods to improve results

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.