Solution of the general moment problem via a one-parameter imbedding

This paper presents a computational theory for the general scalar moment problem. The formalism is sufficiently general to encompass problems in sensor arrays with arbitrary geometry and dynamics, and in nonuniform multidimensional sampling. Given a finite set of moments, the theory provides a test for the existence of a positive measure which is consistent with such data. At the same time, the theory also provides a characterization of all such consistent positive measures. It should be noted that classical results (e.g., in the theory of the trigonometric moment problem, Hamburger, Stieljes, Nevanlinna-Pick interpolation, etc.) are not applicable to the general setting sought herein where there is no natural shift operator in the space spanned by the integration kernels. The centerpiece of the theory is a differential equation which depends on the given finite set of moments and on an arbitrary positive function Ψ-which plays the role of a "free parameter." The differential equation has an exponentially attractive point of equilibrium if and only if there exists a consistent positive measure. For each Ψ, the fixed point determines a corresponding measure. Suitable choice of Ψ allows recovering any measure which is consistent with the data. The fixed point of the differential equation corresponds to an extremum of an entropy-like functional, and the differential equation is constructed via an appropriate homotopy that follows changes in the Lagrange multipliers from a convenient starting value to a value for the multipliers that corresponds to the given moments.