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

Abstract

Let β ≡ β (2n) = {β i } |i| 2n denote a d-dimensional real multisequence, let K denote a closed subset of R d , and let P 2n := {p ∈ R[x 1 , . . . , x d ]: degp 2n}. Corresponding to β, the Riesz functional L ≡ L β : P 2n → R is defined by L( a i x i ) := a i β i . We say that L is K-positive if whenever p ∈ P 2n and p| K 0, then L(p) 0. We prove that β admits a K-representing measure if and only if L β admits a K-positive linear extensionL : P 2n+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.

Solution of the truncated moment problem with variety 𝑦=𝑥³

“…[CF4,Proposition 3.9] together imply that Γ satisfies property (R n,2 ) of [CF8], so the result follows from [CF8,Theorem 1.5]. …”

“…Section 6 is largely independent of the rest of the paper and describes new cases where the truncated moment problem can be solved entirely in terms of positive moment matrix extensions (and without the need for a "flat" extension as in Theorem 1.8). The results of Section 6 are of "intermediate" concreteness, more concrete than the general, but abstract, solutions to the truncated moment problem in Theorem 1.8 or [CF8], but less concrete than Theorem 1.4.…”

“…In [CF8] we studied polynomials p for which the existence of representing measures in the truncated Z p -moment problem can be expressed in terms of moment matrix extensions, without the requirement for an ultimate flat extension as in Theorem 1.8. In the present section we give new examples of this phenomenon.…”

“…Two general, but abstract, solutions to the truncated K-moment problem are known; one involves flat extensions of positive moment matrices [CF6] (cf. Theorem 1.8 below for K = R d ) and the other entails extensions of K-positive linear functionals [CF8] (see Section 6).…”

Positivstellensatz and flat functionals on path ∗-algebras

“…A solution of the K -moment problem is given by the classical Riesz-Haviland theorem. However, the natural analogs of the moment problem and the K -moment problem for truncated moment sequences are more delicate [Fia96,Fia05,Fia08]. We mention the general solution of the truncated K -moment problem in terms of extensions (called the truncated version of the Riesz-Haviland theorem in [Fia91]) and the solution in case of a flat α (see [Fia96]) because of their relevance to the present paper.…”