zbMATH — the first resource for mathematics

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

44A60 Moment problems
26C05 Real polynomials: analytic properties, etc.
12D05 Polynomials in real and complex fields: factorization
Full Text: DOI
[1] Atzmon, A, A moment problem for positive measures on the unit disc, Pacific J. math., 59, 2, 317-325, (1975) · Zbl 0319.44009
[2] Berg, C; Christensen, J.P.R; Jensen, C.U, A remark on the multidimensional moment problem, Math. ann., 243, 163-169, (1979) · Zbl 0416.46003
[3] Berg, C; Maserick, P.H, Polynomially positive definite sequences, Math. ann., 259, 487-495, (1982) · Zbl 0486.44004
[4] Cassier, G, Le problème des moments pour un convexe compact de \(R\)^n, C. R. acad. sci. Paris Sér. I math., 296, 195-197, (1983) · Zbl 0524.44006
[5] Cassier, G, Problème des moments n dimensionnel, mesures quasi-spectrales et semigroupes, Thèse 3ème cycle, (1983), Lyon · Zbl 0521.28007
[6] Fuglede, B, The muldimensional moment problem, Exposition math., 1, 47-65, (1983) · Zbl 0514.44006
[7] Jacobson, N, Lectures in abstract algebra, (1975), Van Nostrand Toronto/New-York/London · Zbl 0314.15001
[8] Lang, S, Algebra, (1965), Addison-Wesley Amsterdam/London/Manila/Singapore/Sydney/Tokyo · Zbl 0193.34701
[9] Maserick, P.H, Moments of measures on convex bodies, Pacific J. math., 68, 135-152, (1977) · Zbl 0323.46054
[10] Mc Gregor, J.L, Solvability criteria for certain N-dimensional moment problems, J. approx. theory, 30, 315-333, (1980) · Zbl 0458.41025
[11] Phelps, R.R, Lectures on Choquet’s theorem, Van nostrand mathematical studies no. 7, (1966), Princeton, N.J. · Zbl 0135.36203
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.