The full infinite dimensional moment problem onsemi-algebraic sets of generalized functions

摘要

We consider ageneric basic semi-algebraic subset S of the space of generalized functions, that is aset given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for aninfinite sequence of generalized functions to be realizable on S, namely to be the moment sequence of a finite measure concentrated on S. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of R~d. In this way, we determine realizability conditions that can be more easily verified than the well-known Haviland type conditions. Our result completely characterizes thesupport of the realizing measure in terms of its moments. As concrete examples of semi-algebraic sets of generalized functions, we consider theset of all Radon measures and theset of all the measures having bounded Radon-Nikodym density w.r.t. the Lebesgue measure.