Moment and Support Sets

• Moment and support sets are mathematical constructs representing sequences of integrals and the underlying domains where nonnegative measures are concentrated.
• Key characterization theorems, including Haviland’s, Schmüdgen’s, and Putinar’s, translate support constraints into algebraic positivity conditions.
• Techniques involving moment matrices, localizing matrices, and flat extension methods enable practical applications in optimization, control, and signal processing.

A moment set is the collection of all sequences or arrays of “moments” that can arise as integrals of monomials (or other prescribed polynomial expressions) against positive measures supported on given sets. Support sets refer to the underlying domains on which these representing measures are concentrated. The interplay between these objects is the core of the $K$-moment problem: characterizing the set of all moment sequences for measures supported on a prescribed subset $K$ of $\mathbb{R}^d$. This domain is central in real algebraic geometry, analysis, and applied fields such as optimization, control, statistics, and signal processing.

1. Formal Definitions: Moment Sets and Support Sets

Let $K \subseteq \mathbb{R}^d$ be a closed set, and let $\mathbb{R}[x] = \mathbb{R}[x_1,\dots,x_d]$ denote the algebra of real polynomials in $d$ variables.

• A linear functional $L: \mathbb{R}[x] \to \mathbb{R}$ is called a $K$-moment functional if there exists a nonnegative Radon measure $\mu$ supported on $K$ so that

$K$0

The measure $K$1 is a representing measure for $K$2.

• The moment sequence associated to $K$3 is $K$4, with multi-index notation: $K$5, where $K$6.
• The moment set of $K$7 is defined as

$K$8

Here, $K$9 denotes the convex cone of all nonnegative Radon measures supported on $\mathbb{R}^d$0 (Amir, 14 Apr 2026).

• The support set in this context is the prescribed set $\mathbb{R}^d$1 on which all representing measures for a given functional (or sequence) are concentrated.

2. Characterization Theorems and Algebraic Structures

Haviland’s Theorem

For a closed $\mathbb{R}^d$2 and linear $\mathbb{R}^d$3, the following are equivalent:

• $\mathbb{R}^d$4 for all $\mathbb{R}^d$5 with $\mathbb{R}^d$6 $\mathbb{R}^d$7 (i.e., $\mathbb{R}^d$8 is nonnegative on the cone $\mathbb{R}^d$9).
• There exists $K \subseteq \mathbb{R}^d$0 such that $K \subseteq \mathbb{R}^d$1 for all $K \subseteq \mathbb{R}^d$2 (Amir, 14 Apr 2026).

Thus, $K \subseteq \mathbb{R}^d$3 is characterized by positivity of the moment functional on polynomials nonnegative on $K \subseteq \mathbb{R}^d$4.

Schmüdgen’s and Putinar’s Positivstellensätze

For compact $K \subseteq \mathbb{R}^d$5, two key algebraic structures arise:

• The preordering $K \subseteq \mathbb{R}^d$6:

$K \subseteq \mathbb{R}^d$7

($K \subseteq \mathbb{R}^d$8 is the cone of sums of squares.) - Schmüdgen’s theorem: Any $K \subseteq \mathbb{R}^d$9 on $\mathbb{R}[x] = \mathbb{R}[x_1,\dots,x_d]$0 belongs to $\mathbb{R}[x] = \mathbb{R}[x_1,\dots,x_d]$1.

• The quadratic module $\mathbb{R}[x] = \mathbb{R}[x_1,\dots,x_d]$2:

$\mathbb{R}[x] = \mathbb{R}[x_1,\dots,x_d]$3

If $\mathbb{R}[x] = \mathbb{R}[x_1,\dots,x_d]$4 is Archimedean (i.e., $\mathbb{R}[x] = \mathbb{R}[x_1,\dots,x_d]$5 such that $\mathbb{R}[x] = \mathbb{R}[x_1,\dots,x_d]$6), then by Putinar's theorem, any $\mathbb{R}[x] = \mathbb{R}[x_1,\dots,x_d]$7 on $\mathbb{R}[x] = \mathbb{R}[x_1,\dots,x_d]$8 lies in $\mathbb{R}[x] = \mathbb{R}[x_1,\dots,x_d]$9. Under Archimedeanity, nonnegativity of $d$0 on $d$1 suffices for $d$2 to be a $d$3-moment functional (Amir, 14 Apr 2026, Schmüdgen, 2023).

These results translate geometric support constraints into purely algebraic positivity requirements.

3. Moment Matrices, Localizing Matrices, and Flat Extension

Given a (truncated or full) moment sequence $d$4, construct for each $d$5 the moment matrix $d$6 indexed by multi-indices $d$7:

$d$8

A necessary condition for $d$9 is that $L: \mathbb{R}[x] \to \mathbb{R}$0 for every $L: \mathbb{R}[x] \to \mathbb{R}$1.

For each defining polynomial $L: \mathbb{R}[x] \to \mathbb{R}$2, the localizing matrix $L: \mathbb{R}[x] \to \mathbb{R}$3 is

$L: \mathbb{R}[x] \to \mathbb{R}$4

The truncated $L: \mathbb{R}[x] \to \mathbb{R}$5-moment problem considers whether a given $L: \mathbb{R}[x] \to \mathbb{R}$6 is the sequence of moments for some measure $L: \mathbb{R}[x] \to \mathbb{R}$7 supported on $L: \mathbb{R}[x] \to \mathbb{R}$8. The Curto–Fialkow flat extension theorem states: if $L: \mathbb{R}[x] \to \mathbb{R}$9 and a higher-order extension $K$0 are PSD and satisfy

$K$1

then a finitely atomic representing $K$2 on $K$3 exists, with number of atoms equal to $K$4 (Amir, 14 Apr 2026, Schmüdgen, 2023).

4. Extraction of Support from Moments

The support of a measure representing a moment sequence is determined by the annihilating kernel of the moment matrix. For a full moment sequence $K$5 with representing measure $K$6, any polynomial $K$7 in the kernel of $K$8 satisfies:

$K$9

Therefore,

$\mu$0

When the flat extension property holds, the vanishing locus of the kernel is finite and the atoms of the representing measure are exactly the common real zeros of the polynomials in $\mu$1 (Amir, 14 Apr 2026).

5. Generalizations: Infinite Dimensions, Function Spaces, and Specialized Supports

The moment-support machinery extends to several advanced settings:

• Infinite-dimensional spaces: The moment problem for measures on general nuclear spaces or for random measures involves representing support sets $\mu$2 as generalized basic closed semi-algebraic sets defined by countable systems of polynomial inequalities. The positivity requirements for the Riesz functional, and the encoding of support information through polynomial constraints, remain conceptually identical [(Infusino et al., 2018); (Infusino et al., 2013)].
• Function spaces: In the setting of Schwartz functions $\mu$3 and Gelfand–Shilov spaces, every sequence is a $\mu$4-moment sequence (moment-functional supported in $\mu$5) if and only if certain spaces of polynomials with boundary-weighted growth control are finite-dimensional. Regularity (thickness) of $\mu$6 at infinity becomes necessary for solvability in these smooth function classes (Debrouwere, 20 May 2025).
• Stieltjes moment sequences with shifted supports: A sequence $\mu$7 is a $\mu$8-Stieltjes moment sequence (i.e., has a representing measure supported in $\mu$9) if and only if its generating function admits a continued-fraction expansion with coefficients satisfying certain inequalities involving $K$0 (Sokal et al., 2024).
• Support recovery from partial or marginal moments: Algorithms based on semidefinite programming, moment matrix pencils, or Christoffel functions can efficiently compute enclosing intervals, approximate support sets, or extract the underlying support of a measure from partial (or marginal) moment information [(Jasour et al., 2014); (Lasserre, 2010); (Pauwels et al., 2018)].

6. Applications: Optimization, Identification, and Robustness

Moment and support sets underpin diverse domains:

• Polynomial and semi-infinite programming: Many optimization problems over measures with moment and support constraints are naturally reformulated as linear conic problems over moment cones, with tractable semidefinite relaxations via Moment–SOS techniques. Feasibility, optimality, and extraction of supporting points rely fundamentally on the interplay between moments and support sets (Nie et al., 2021, Huang et al., 2022, Hu et al., 2024).
• Economic modeling and identification: In incomplete models, structural or counterfactual identification is closely tied to properties of support and moment closures. Support-function approaches yield sharp characterizations for identified sets and their closures, even when traditional random set bounds fail due to unboundedness or lack of finite support (Li, 8 Mar 2026).
• Data analysis and learning: Empirical moment matrices encode zeros and algebraic structure of the support, enabling support reconstruction and density estimation in high-dimensional data, especially when the underlying support is a real algebraic variety or a sparse set (Pauwels et al., 2018, Henrion et al., 2020).

7. Summary Table: Central Mathematical Objects

Object	Definition	Role in Moment/Support Theory
$K$1 (moment set)	Sequences arising as $K$2	Characterizes realizable moment sequences
Support set $K$3	Closed subset of $K$4	Prescribes possible locations for measures
Moment matrix $K$5	$K$6	Translates moment positivity to SDP
Quadratic module $K$7, preordering $K$8	Algebraic positivity cones	Encodes support via polynomial inequalities
Localizing matrix $K$9	For $K$00, entries $K$01	Enforces support constraints positivity
Flat extension property	Rank condition on $K$02 and $K$03	Ensures atomic measure, support extraction
Kernel of $K$04	Polynomials vanishing $K$05-a.e.	Annihilator; determines support algebraically

These frameworks and tools interlock: support sets $K$06 restrict possible representing measures, while moment sets capture integrability and positivity data. The algebraic-geometric machinery (quadratic modules, moment matrices, preorderings) enables both advanced theoretical results and efficient computational algorithms for support identification and recovery.

References: (Amir, 14 Apr 2026, Schmüdgen, 2023, Infusino et al., 2018, Sokal et al., 2024, Jasour et al., 2014, Lasserre, 2010, Debrouwere, 20 May 2025, Infusino et al., 2013, Huang et al., 2022, Henrion et al., 2020, Pauwels et al., 2018, Nie et al., 2021, Hu et al., 2024, Kimsey et al., 2019, Li, 8 Mar 2026).