[Paper Review] A Riesz-Haviland type result for truncated moment problems with solutions in $L^1$

Călin-Grigore Ambrozie|arXiv (Cornell University)|Nov 28, 2011

Numerical methods in inverse problems14 references4 citations

TL;DR

This paper establishes a Riesz-Haviland-type characterization for truncated moment problems with representing densities in $L^1(T, dt)$, showing that such densities exist if and only if the associated Riesz functional is strictly positive on nonnegative polynomials over a regular closed set $T$. The key contribution is identifying the dense interior of the moment cone as the set of data admitting $L^1$-density representing measures under regularity conditions on the support.

ABSTRACT

We give a version of the Riesz-Haviland theorem for truncated moments problems, characterizing the existence of the representing measures that are absolutely continuous with respect to the Lebesgue measure. The existence of such representing densities describes the dense interior of the convex cone of all data having nonnegative Borel representing measures. A natural regularity assumption on the support is required.

Motivation & Objective

• To characterize the existence of representing measures that are absolutely continuous with respect to Lebesgue measure in truncated moment problems.
• To identify the conditions under which the moments of a nonnegative $L^1$ density on a regular closed set $T\subset\mathbb{R}^n$ yield a solution to the truncated moment problem.
• To show that the set of moment sequences admitting $L^1$-density solutions forms the dense interior of the cone of all moment sequences with nonnegative Borel representing measures.
• To provide a characterization independent of prior results by Curto and Fialkow, relying instead on functional analytic tools and the classical Riesz-Haviland theorem.

Proposed method

• Introduces a notion of regularity for closed sets $T\subset\mathbb{R}^n$, requiring positive Lebesgue measure in every neighborhood of each point in $T$, ensuring topological and measure-theoretic regularity.
• Defines a regular index set $I\subset\mathbb{Z}_+^n$ closed under coordinate-wise 0-1 combinations, ensuring compatibility with moment functionals.
• Uses the Riesz-Haviland theorem as a foundation, extending it to the truncated case by analyzing strict positivity of the Riesz functional $\varphi_g$ on nonnegative polynomials over $T$.
• Applies Hahn-Banach and Krein-Mazur type arguments to extend functionals while preserving strict positivity, enabling construction of global positive functionals.
• Establishes that strict positivity $\varphi_g p > 0$ for all nonzero $p\in P_I$ with $p(t)\geq 0$ on $T$ implies local uniform positivity, leading to a neighborhood of $g$ in the moment space where the same condition holds.
• Uses compactness and duality to show that strict positivity implies a lower bound $\varphi_g p \geq c\|p\|$ on the cone of nonnegative polynomials, enabling iterative extension to full moment sequences.

Experimental results

Research questions

• RQ1 Under what conditions does a truncated moment sequence admit a representing density in $L^1(T, dt)$ for a regular closed set $T\subset\mathbb{R}^n$?
• RQ2 How can the existence of $L^1$-density solutions be characterized in terms of the positivity properties of the associated Riesz functional?
• RQ3 What is the topological structure of the set of moment sequences admitting $L^1$-density solutions within the larger cone of all nonnegative Borel representing measures?
• RQ4 Can the characterization of $L^1$-density solutions be derived independently of the Curto-Fialkow extension theorem for truncated moment problems?

Key findings

• A truncated moment sequence $g=(g_i)_{i\in I}$ admits a representing density $f\in L^1_+(T, dt)$ if and only if the associated Riesz functional $\varphi_g$ satisfies $\varphi_g p > 0$ for all nonzero polynomials $p\in P_I$ such that $p(t)\geq 0$ for all $t\in T$.
• The set of moment sequences admitting $L^1$-density solutions is the dense interior of the convex cone of all moment sequences with nonnegative Borel representing measures on $T$.
• The strict positivity condition $\varphi_g p > 0$ for all nonzero nonnegative $p$ on $T$ is equivalent to the existence of a uniform lower bound $\varphi_g p \geq c\|p\|$ on the unit sphere of nonnegative polynomials in $P_I$.
• The proof constructs a sequence of positive extensions of $\varphi_g$ to higher-degree polynomial spaces, ultimately yielding a positive linear functional on $\mathbb{R}[X_1,\dots,X_n]$ that satisfies the Riesz-Haviland criterion.
• The existence of such a functional guarantees the existence of a nonnegative Borel measure $\nu$ on $T$ with moments matching $g$, and by the strict positivity and $L^1$-regularity, $\nu$ is absolutely continuous with respect to Lebesgue measure.
• The result holds under the assumption that $T$ is regular, meaning every point in $T$ is a Lebesgue density point, ensuring the support has positive measure in every neighborhood.