Problème des moments sur un compact de \({\mathbb{R}}^ n\) et décomposition de polynômes a plusieurs variables
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, \textit{Ch. Berg} and \textit{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 \textit{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.)
scientific article; zbMATH DE number 3886535