In this article we show that measures, with finite support on the real line, are the unique solution to an algorithm, named support pursuit, involving only a finite number of generalized moments (which encompass the standard moments, the Laplace transform, the Stieljes transformation, etc...). The support pursuit share related geometric properties with basis pursuit of Chen, Donoho and Saunders. As a matter of fact we extend some standard results of compressed sensing (the dual polynomial, the nullspace property) to the signed measure framework. We express exact reconstruction in terms of a simple interpolation problem. We prove that every nonnegative measure, supported by a set containing s points, can be exactly recovered from only 2s+1 generalized moments. This result leads to a new construction of deterministic sensing matrices for compressed sensing. In particular, we prove that one can recover all nonnegative s-sparse vectors from only 2s+1 linear measurements.
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