[Paper Review] Multivariate Determinateness

Mihai Putinar, Konrad Schmüdgen|ArXiv.org|Oct 5, 2008

Advanced Statistical Methods and Models14 references4 citations

TL;DR

This paper provides a comprehensive analysis of the multivariate moment problem's determinacy, unifying approaches from Hilbert space operators, complex analysis, polynomial approximation, and integral geometry. It establishes new determinacy criteria, particularly proving that moment problems on virtually compact sets in real algebraic curves are determinate, extending classical one-dimensional results to higher dimensions with geometric and algebraic conditions.

ABSTRACT

The uniqueness question of the multivariate moment problem is studied by different methods: Hilbert space operators, complex function theory, polynomial approximation, disintegration, integral geometry. Most of the known results in the multi-dimensional case are reviewed and reproved, and a number of new determinacy criteria are developed.

Motivation & Objective

• To resolve the long-standing open problem of determining uniqueness conditions for multivariate moment problems, which remain less understood than in the one-dimensional case.
• To unify and re-derive known determinacy criteria using diverse mathematical frameworks: operator theory, complex function theory, and algebraic geometry.
• To introduce and prove new determinacy conditions based on the concept of 'virtually compact' sets in real algebraic curves.
• To extend classical results—such as Riesz's density criterion and Carleman's quasi-analyticity condition—to the multivariate setting with geometric constraints.
• To provide a self-contained, systematic exposition of the state-of-the-art in multivariate determinateness, including new parametrizations and criteria.

Proposed method

• Utilizes Hilbert space operators and spectral theory, particularly the resolvent identity for self-adjoint extensions of Jacobi matrices, to characterize solutions to moment problems.
• Applies complex analytic methods, including Cauchy and Laplace transforms, to derive uniqueness via quasi-analyticity of analytic continuations.
• Employs integral geometry and disintegration techniques to reduce dimensionality and analyze measures supported on convex wedges.
• Introduces the concept of 'virtually compact' sets in real algebraic curves, defined via the behavior of points at infinity in the projective closure.
• Uses algebraic geometry tools to define the ring of bounded regular functions on a set, linking it to determinacy via separation of points.
• Applies the theory of orthogonal polynomials and continued fractions in higher dimensions, generalizing one-dimensional spectral methods.

Experimental results

Research questions

• RQ1 Under what geometric and algebraic conditions on a closed subset $ K \subset \mathbb{R}^n $ is the multivariate moment problem determinate?
• RQ2 How can the classical one-dimensional determinacy criteria—such as Carleman’s condition or Riesz’s density criterion—be generalized to the multivariate case?
• RQ3 What role does the topology and geometry of the support of a measure play in determining uniqueness of the representing measure?
• RQ4 Can the concept of 'virtually compact' sets in real algebraic curves serve as a unifying criterion for multivariate determinateness?
• RQ5 To what extent can integral geometry and disintegration methods reduce the multivariate moment problem to lower-dimensional or solvable cases?

Key findings

• A moment problem on a closed subset $ K \subset X(\mathbb{R}) $, where $ X $ is an irreducible real algebraic curve, is determinate if $ K $ is virtually compact.
• The ring $ \mathcal{H}(K) $ of bounded regular functions on $ K $ separates points in $ X(\mathbb{R}) $ if and only if $ K $ is virtually compact, which implies determinacy.
• For any non-constant $ f \in \mathbb{R}[X] $ with $ f \geq 1 $ on $ K $, if $ f p_n \to 1 $ in the $ L^2(K) $-norm, then the moment problem on $ K $ is determinate.
• The condition of virtual compactness holds, for example, when the leading form of the defining polynomial is not a product of real linear factors, or when singular points at infinity introduce non-real components.
• Examples include hyperelliptic curves $ y^2 = q(x) $ with $ \deg(q) $ divisible by 4, where determinacy holds if $ y $ is bounded above or below on the set.
• The paper establishes that moment problems on sets like $ \{(x,t) \in K_1 \times \mathbb{R} \mid t f(x) = 1\} $ are determinate, even when $ K $ is non-compact, due to the structure of $ \mathcal{H}(K) $.