Entropy-convergence, Instability in Stieltjes and Hamburger Moment Problems

The recovering of a positive density, of which a nite number of moments is assigned, is considered (in the Stieltjes and Hamburger moment problems). In the choice of the approximant the Maximum Entropy approach is adopted. Two main problems are taken into account. 1. A review of diierent criteria concerning the determinacy and indeterminacy of the innnite moment problem is presented. This review includes criteria based on the moments themselves or on the coeecients of the three-terms recurrence relation characterizing the orthogonal polynomials arising from the moment sequence. A particular criterion, guaranteeing the determinacy of the moment problem, is endowed with a geometrical interpretation, which allows the proof of entropy convergence. More precisely, it is proved that, whenever the sequence of assigned moments arises a determinate moment problem, then the approximants converge in entropy to the recovering function. 2. The entropy convergence allows us to employ a large number of moments. Unstability of the recovering is proved when an increasing number of moments is used. Subsequently the condition number, the ratio between the relative error of the function and the relative error of the modiied moment, is analyzed. We also prove that the condition number is related to the orthogonal polynomials generated by the sequence of moments.