Fractional Moments and Maximum Entropy: Geometric Meaning
We consider the problem of recovering a probability density on a bounded or unbounded subset D of [0, ∞), from the knowledge of its sequence of fractional moments within a maximum entropy (MaxEnt) setup. Based upon entropy convergence results previously formulated, the fractional moments are selected so that the entropy of the MaxEnt approximation be minimum. A geometric interpretation of the reconstruction procedure is formulated as follows: the two moment curves generated by the unknown density and its MaxEnt approximation are interpolating in Hermite-Birkoff sense; that is, they are both interpolating and tangent at the selected nodes.