Hausdorff moment problem via fractional moments
  We outline an efficient method for the reconstruction of a probability density function from the knowledge of its infinite sequence of ordinary moments. The approximate density is obtained resorting to maximum entropy technique, under the constraint of some fractional moments. The latter ones are obtained explicitly in terms of the infinite sequence of given ordinary moments. It is proved that the approximate density converges in entropy to the underlying density, so that it demonstrates to be useful for calculating expected values. 
Pierluigi Novi Inverardi, Alberto Petri, Giorgio Pontuale and Aldo   Tagliani