An analogue of the Riesz-Haviland theorem for the truncated moment problem.(English)Zbl 1158.44003

Every $$d$$-dimensional real multisequence $$\beta ^{(\infty )}=(\beta _{i})_{i\in \mathbb{N}^{d}}$$ can be seen as a linear operator $$L_{\beta ^{(\infty )}}:\mathcal{P}\rightarrow \mathbb{R}$$ acting on the space $$\mathcal{P=}\mathbb{R}[x_{1},\ldots ,x_{d}]$$ of real polynomials in $$\mathbb{R}^{d}$$ by setting $$L_{\beta ^{(\infty )}}(x^{i})=\beta _{i}$$ (where $$x^{i}:=x_{1}^{i_{1}}\cdots x_{d}^{i_{d}}$$). Given a closed subset $$K$$ of $$\mathbb{R}^{d}$$ the Haviland generalization of the Riesz theorem asserts that $$L_{\beta ^{(\infty )}}$$ can be represented by a measure $$\mu$$ supported in $$K$$ (that is, $$L_{\beta^{(\infty)}}(p)=\int p\,d\mu$$ for all $$p\in \mathcal{P}$$) if, and only if, $$L_{\beta ^{(\infty )}}$$ is $$K$$-positive (that is, $$L_{\beta ^{(\infty )}}(p)\geq 0$$ whenever the polynomial $$p$$ is positive on $$K$$). The truncated moment problem concerns the case where the data are restricted to $$\beta ^{(2n)}=(\beta _{i})_{i\in \mathbb{N}^{d},|i|\leq 2n}$$ defining thus a linear operator $$L_{\beta ^{(2n)}}:\mathcal{P}_{2n}\rightarrow \mathbb{R}$$, where $$\mathcal{P}_{2n}$$ stands for the space of polynomials whose degree is less or equal to $$2n$$. In this context, a similar representation result is true whenever $$K$$ is compact: $$L_{\beta ^{(2n)}}$$ admits a $$K$$-representing measure if and only if it is $$K$$-positive (that is, $$L_{\beta ^{(2n)}}(p)\geq 0$$ for all polynomials of $$\mathcal{P}_{2n}$$ that are positive on $$K$$). The authors present an example (Example 2.1) where the sufficiency fails if the compactness condition is removed. They subsequently establish the following criterium (Theorem 2.2):
$$L_{\beta ^{(2n)}}$$ admits a $$K$$-representing measure if and only if it admits a $$K$$-positive linear extension $$\widetilde{L}:\mathcal{P}_{2n+2}\rightarrow \mathbb{R}$$. They also discuss the particular case of a closed semialgebraic set $$S=\{x\in \mathbb{R }^{d}:q_{i}(x)\geq 0,\{q_{i}\}_{i\in \{1,\ldots ,m\}}\in \mathcal{P}\}$$. In such a case they show that $$S$$ solves the truncated moment problem in terms of natural degree-bounded positivity conditions if and only if every polynomial $$p$$ that is strictly positive on $$S$$ admits a degree-bounded weighted SOS-representation (in terms of the polynomails $$q_{i}$$ defining $$S$$).

MSC:

 44A60 Moment problems 47A57 Linear operator methods in interpolation, moment and extension problems 14P10 Semialgebraic sets and related spaces

Software:

GloptiPoly; SeDuMi
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References:

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