Maximum entropy and moment problems.

Abstract

We study moment problems in which one searches for a probability density on ${\R}^n$ by using insufficient information given in an integral form. We characterize the existence of the representing densities for a finite multi--sequence of moments by the solvability of a concrete finite system of equations. Its solution provides the (unique) representing density of maximum entropy allowed by the given data, that turns out to be the exponential of a polynomial to be determined. For all densities of this form, the system to be solved can be taken linear to be provided sufficiently many moments are known.