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The problem of moments on compact semi-algebraic sets. (Problème des moments sur les compacts semi-algébriques.) (French. Abridged English version) Zbl 0857.44004

Summary: Let \(K=\{x\in \mathbb{R}^n;\;p_1(x)\geq 0, \dots, p_N(x) \geq 0\}\) be a compact semi-algebraic subset of \(\mathbb{R}^n\), where \(p_1, \dots, p_N\), are polynomials normalized by \(|p_j |_{\infty,K} \leq 1\) \((1\leq j \leq N,\;N \geq n+1)\), and such that \(p_1, \dots, p_n\) are of degree one and linearly independent. Then the problem of moments \[ a_\alpha = \int_K x^\alpha d\mu(x), \quad \alpha\in\mathbb{N}^n, \] has as a solution a positive Borel measure \(\mu\) on \(K\) if and only if the associated functional \(L\in \mathbb{R}[x]'\), \(L(x^\alpha) = a_\alpha (\alpha\in \mathbb{N}^n)\), is nonnegative on the set of polynomials of the form \[ p_1^{m_1} \cdots p_N^{m_N} (1-p_1)^{k_1} \cdots (1-p_N)^{k_N}, \] where \(m_1, \dots, m_N\), \(k_1, \dots, k_N\) are arbitrary nonnegative integers.

MSC:

44A60 Moment problems
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