On the convergence of moment problems

We study the problem of estimating a nonnegative density, given a finite number of moments. Such problems arise in numerous practical applications. As the number of moments increases, the estimates will always converge weak * as measures, but need not converge weakly in L₁. This is related to the existence of functions on a compact metric space which are not essentially Riemann integrable (in some suitable sense). We characterize the type of weak convergence we can expect in terms of Riemann integrability, and in some cases give error bounds. When the estimates are chosen to minimize an objective function with weakly compact level sets (such as the Bolzmann-Shannon entropy) they will converge weakly in L₁. When an L_p norm (1 < p < ∞) is used as the objective, the estimates actually converge in norm. These results provide theoretical support to the growing popularity of such methods in practice.