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Problème des moments sur un compact de $${\mathbb{R}}^ n$$ et décomposition de polynômes a plusieurs variables. (French) Zbl 0556.44006
Given a positive measure $$\mu$$ on a compact set K, one has an associated sequence of moments $$a_ k=\int_{k}t^ kd\mu (t),$$ with the usual meaning of $$t^ k=t_ 1^{k_ 1}...t_ n^{k_ n},k\in {\mathbb{N}}^ n.$$ The moment problem is to characterize the sequences obtained in this way. It is understood in dimension 1, where, for example, Ch. Berg and P. H. Maserick [Math. Ann. 259, 487-495 (1982; Zbl 0486.44004)] proved that if $$K=P^{-1}({\mathbb{R}})$$, then $$(s_ k)$$ is a sequence of moments of K iff $$(s_ k)$$ and $$P(s)_ k$$ are sequences of positive type. For $$n>1$$ they were able to prove the equivalence only if $$(s_ k)$$ and $$(P(s))_ k$$ are a sequence of moments, and posed the question of whether the theorem remains true when the sequences $$(s_ k)$$ and $$(P(s))_ k$$ satisfy the weaker condition of being of positive type.
The author shows that this is true for polynomials P of a special type; either for a polynomial in two variables whose homogeneous part of highest degree is strictly negative on $${\mathbb{R}}^ 2\setminus \{0\}$$, or for polynomials whose homogeneous part of highest degree is of the form $$-(\alpha_ 1X_ 1^{2p}+...+\alpha_ nX_ n^{2p})-H_{2p},$$ with $$p\in {\mathbb{N}}$$, $$\alpha_ k\in {\mathbb{R}}^*_+$$ and $$H_{2p}^ a$$sum of squares. He also applies his method to solve the moment problem for p- balls, $$K=\{| x_ 1|^ p+...+| x_ n|^ p\leq 1\},$$ generalizing a result for $$p=2$$ by J. L. McGregor [J. Approximation Theory 30, 315-333 (1980; Zbl 0458.41025)]. He concludes by giving a decomposition theorem for strictly positive polynomials on $$K=R^{- 1}({\mathbb{R}}_+)$$ with R one of the previously cited two types. The key to his results is a functional analytic characterization of a necessary and sufficient condition for a moment sequence. Given $$(a_ k| k\in {\mathbb{N}}^ n)$$, and $$T=\sum_{r}u_ rX^ r$$ a polynomial, let $$M_ T(a)=[m_{ij}],$$ where $$m_{ij}=\sum u_ ra_{i+j+r}.$$ Then $$(a_ k)$$ is a moment sequence for K iff $$M_ T(a)$$ defines operators of positive type when T runs over a suitable set of polynomials depending on K (for whose definition one must refer to the paper.)
Reviewer: R.Johnson

##### MSC:
 44A60 Moment problems 26C05 Real polynomials: analytic properties, etc. 12D05 Polynomials in real and complex fields: factorization
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##### References:
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