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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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