On the sufficiency of [formula omitted]-positivity for truncated compactly supported generalized moment problems

Abstract

Given a compact set K and a finite set of continuous basis functions, the truncated generalized K-moment problem asks for a characterization of all sequences that can be obtained as moments, with respect to the basis functions, of some nonnegative measure with support in K. Under a condition that a certain convex cone is nonempty, the moment sequences can be characterized as being the elements in the dual cone of the closure. This dual cone is also known as the set of all sequences for which the corresponding Riesz functional is K-positive. Here, we give a short, alternative proof of this statement, based on convex optimization and duality. We then present two examples. The first example shows that if the nonemptiness condition is removed, then K-positivity is in general no longer a sufficient condition for a sequence to be a moment sequence. Nevertheless, the second example shows that there are moment problems where the convex cone is empty, but for which K-positivity is still a necessary and sufficient condition for the existence of a representing measure.