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 . 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 . 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.