An analogue of the Riesz–Haviland theorem for the truncated moment problem

Let β ≡ β(2n) = {βi }|i| 2n denote a d-dimensional real multisequence, let K denote a closed subset of Rd, and let P2n := {p ∈ R[x1, . . . , xd ]: degp 2n}. Corresponding to β, the Riesz functional L ≡ Lβ : P2n →R is defined by L( aixi ) := aiβi . We say that L is K-positive if whenever p ∈ P2n and p| K 0, then L(p) 0. We prove that β admits a K-representing measure if and only if Lβ admits a K-positive linear extension ˜L :P2n+2→R. This provides a generalization (from the full moment problem to the truncated moment problem) of the Riesz–Haviland theorem. We also show that a semialgebraic set solves the truncated moment problem in terms of natural “degree-bounded” positivity conditions if and only if each polynomial strictly positive on that set admits a degree-bounded weighted sum-of-squares representation.