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)].


44A60 Moment problems
14P10 Semialgebraic sets and related spaces


Zbl 0636.44007
Full Text: DOI EuDML


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