Sets of superresolution and the maximum entropy method on the mean

Consider the problem of recovering a probability measure supported on a compact polish space U, endowed with a probability P, when the available measurements only concern some of its Phi-moments; Phi is here a given k-dimensional continuous real function on U. Provided the true moment c lies on the boundary of the convex hull of Phi(U), we exhibit a support of uniform concentration, that is, a measurable set R(c,delta)(epsilon) (depending on a small positive number delta) such that for any solution mu which satisfies parallel to integral Phi d mu-c parallel to(2) less than or equal to epsilon, we have mu(R(c,delta)(epsilon)) greater than or equal to 1 - K(epsilon) and P(R(c,delta)(epsilon)) less than or equal to C.K(epsilon), where K(epsilon) decreases to 0 with epsilon and C is a constant number. The construction of R(c,delta)(epsilon) and the results are intimately connected with the maximum entropy method on the mean (MEM) developed by Gamboa and Gassiat. This method gives a general framework for superresolution theory via Pythagoras inequalities on families of dissimilarities linked with MEM. In particular cases, we prove that K(epsilon) is the exact rate of uniform concentration over R(c,delta)(epsilon).