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The $$K$$-moment problem for compact semi-algebraic sets. (English) Zbl 0744.44008
Suppose $$K$$ is a closed subset of $$\mathbb{R}^ d$$. A function $$f: \mathbb{N}^ d_ 0\to R$$ is called a $$K$$-moment sequences if there exists a positive Borel measure $$\mu\in M(\mathbb{R}^ d)$$ supported by $$K$$ such that $$f(\alpha)$$ is the $$\alpha$$-th moment of $$\mu$$, i.e., $$f(\alpha)=\int x^ \alpha d\mu$$, $$\forall\alpha\in\mathbb{N}^ d_ 0$$. The main result of this note characterizes the $$K$$-moment sequences for compact semi- algebraic sets $$K$$. Theorem 1 subsumes the above and proves a conjecture of Berg and Maserick [see C. Berg, Moments in mathematics. AMS Short Course, San Antonia/Tex. 1987, Proc. Symp. Appl. Math. 37, 110-124 (1987; Zbl 0636.44007)].

##### MSC:
 44A60 Moment problems 14P10 Semialgebraic sets and related spaces
##### Keywords:
$$K$$-moment problem; compact semi-algebraic sets
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##### References:
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