Solving moment problems by dimensional extension. (English) Zbl 0939.44003

Putinar, Mihai; Vasilescu, Florian-Horia

Ann. Math. (2) 149, No. 3, 1087-1107 (1999).

From the introduction: The first part of this paper is devoted to an analysis of moment problems in \(\mathbb{R}^n\), \(n\geq 1\), with supports contained in a closed set defined by finitely many polynomial inequalities. The second part of the paper uses the representation results of positive functionals on certain spaces of rational functions developed in the first part, for decomposing a polynomial which is positive on such a semi-algebraic set into a canonical sum of squares of rational functions times explicit multipliers.
The present paper starts from an idea, about solving moment problems by a change of basis via an embedding of \(\mathbb{R}^n\) into a submanifold of a higher dimensional Euclidean space. We prove that certain \((n+1)\)-dimensional extensions of a moment sequence are naturally characterized by positivity conditions and moreover, these extensions parametrize all possible solutions of the moment problem.

Reviewer: A.L.Brodskij (Severodonetsk)

MSC:

44A60	Moment problems
46C05	Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

Keywords:

dimensional extension; Hilbert space; \(\mathbb{C}\)-algebra; moment problems; polynomial inequalities; positive functionals