Two-Dimensional Moment Problem

• The two-dimensional moment problem is defined as determining whether a bivariate sequence can be represented as the moments of a positive measure on ℝ² or a prescribed support set K.
• It integrates methods from analysis, real algebraic geometry, operator theory, convex geometry, and semidefinite optimization to establish conditions for measure representation.
• Truncated formulations employ moment and localizing matrices with flat extension criteria to ensure finite atomic representing measures and practical computational frameworks.

The two-dimensional moment problem asks when a family of real numbers indexed by $\mathbb{N}^2$ is representable as moments of a positive measure on $\mathbb{R}^2$, either on the full plane or on a prescribed support set $K\subseteq \mathbb{R}^2$. In its basic form, one seeks a positive Radon or Borel measure $\mu$ such that

$$s_{\alpha}=\int_{\mathbb{R}^2} x_1^{\alpha_1}x_2^{\alpha_2}\,d\mu(x_1,x_2),\qquad \alpha=(\alpha_1,\alpha_2)\in\mathbb{N}^2.$$

The corresponding $K$-moment problem requires $\operatorname{supp}\mu\subseteq K$. In two variables, the subject lies at the intersection of analysis, real algebraic geometry, operator theory, convex geometry, and semidefinite optimization, and it includes both full and truncated formulations (Amir, 14 Apr 2026).

1. Classical formulation and positivity

A two-dimensional moment sequence is a family $s=(s_\alpha)_{\alpha\in\mathbb{N}^2}$, and it defines a linear functional $L:\mathbb{R}[x_1,x_2]\to\mathbb{R}$ by

$$L(p)=\sum_{\alpha} p_\alpha s_\alpha=\int_{\mathbb{R}^2} p(x_1,x_2)\,d\mu(x_1,x_2),$$

for $\mathbb{R}^2$0. The central existence statement is Haviland’s theorem in dimension two: if $\mathbb{R}^2$1 is closed, then $\mathbb{R}^2$2 admits a representing positive Radon measure supported on $\mathbb{R}^2$3 if and only if $\mathbb{R}^2$4 for every polynomial $\mathbb{R}^2$5 with $\mathbb{R}^2$6. For $\mathbb{R}^2$7, this becomes positivity on the cone $\mathbb{R}^2$8 (Amir, 14 Apr 2026).

In truncated form, only finitely many moments are prescribed. For degree $\mathbb{R}^2$9, one asks for $K\subseteq \mathbb{R}^2$0 such that

$K\subseteq \mathbb{R}^2$1

with $K\subseteq \mathbb{R}^2$2. The truncated moment matrix $K\subseteq \mathbb{R}^2$3, indexed by $K\subseteq \mathbb{R}^2$4, is given by

$K\subseteq \mathbb{R}^2$5

Its positivity is equivalent to $K\subseteq \mathbb{R}^2$6 for all polynomials $K\subseteq \mathbb{R}^2$7 of degree $K\subseteq \mathbb{R}^2$8 (Kimsey et al., 2019).

A persistent structural distinction from one dimension is that the positivity of a single Hankel or Toeplitz object does not generally solve the support-constrained problem in higher dimensions. One-dimensional Hamburger and circular moment problems admit such criteria, whereas in two dimensions “except some fortunate configurations, in higher dimensions there are no non-negativity criteria for the power moments of a measure to be supported by a prescribed closed set” (Kimsey et al., 2019).

2. Semialgebraic supports, quadratic modules, and positivity certificates

When the support is a basic closed semialgebraic set

$K\subseteq \mathbb{R}^2$9

the analytic cone of polynomials nonnegative on $\mu$0 is approximated algebraically by two standard cones. The quadratic module is

$\mu$1

and the preordering is

$\mu$2

One always has $\mu$3 (Amir, 14 Apr 2026).

Two Positivstellensätze govern the compact case. Schmüdgen’s theorem states that if $\mu$4 is compact and $\mu$5 on $\mu$6, then $\mu$7. Putinar’s theorem states that if $\mu$8 is Archimedean, meaning there exists $\mu$9 such that

$$s_{\alpha}=\int_{\mathbb{R}^2} x_1^{\alpha_1}x_2^{\alpha_2}\,d\mu(x_1,x_2),\qquad \alpha=(\alpha_1,\alpha_2)\in\mathbb{N}^2.$$0

then $$s_{\alpha}=\int_{\mathbb{R}^2} x_1^{\alpha_1}x_2^{\alpha_2}\,d\mu(x_1,x_2),\qquad \alpha=(\alpha_1,\alpha_2)\in\mathbb{N}^2.$$1 on $$s_{\alpha}=\int_{\mathbb{R}^2} x_1^{\alpha_1}x_2^{\alpha_2}\,d\mu(x_1,x_2),\qquad \alpha=(\alpha_1,\alpha_2)\in\mathbb{N}^2.$$2 implies $$s_{\alpha}=\int_{\mathbb{R}^2} x_1^{\alpha_1}x_2^{\alpha_2}\,d\mu(x_1,x_2),\qquad \alpha=(\alpha_1,\alpha_2)\in\mathbb{N}^2.$$3. The Archimedean condition is the “algebraic shadow of compactness”: it implies $$s_{\alpha}=\int_{\mathbb{R}^2} x_1^{\alpha_1}x_2^{\alpha_2}\,d\mu(x_1,x_2),\qquad \alpha=(\alpha_1,\alpha_2)\in\mathbb{N}^2.$$4, hence compactness. The preordering $$s_{\alpha}=\int_{\mathbb{R}^2} x_1^{\alpha_1}x_2^{\alpha_2}\,d\mu(x_1,x_2),\qquad \alpha=(\alpha_1,\alpha_2)\in\mathbb{N}^2.$$5 is always Archimedean when $$s_{\alpha}=\int_{\mathbb{R}^2} x_1^{\alpha_1}x_2^{\alpha_2}\,d\mu(x_1,x_2),\qquad \alpha=(\alpha_1,\alpha_2)\in\mathbb{N}^2.$$6 is compact, whereas $$s_{\alpha}=\int_{\mathbb{R}^2} x_1^{\alpha_1}x_2^{\alpha_2}\,d\mu(x_1,x_2),\qquad \alpha=(\alpha_1,\alpha_2)\in\mathbb{N}^2.$$7 may fail to be Archimedean unless the generators already include a coordinate bound such as a ball constraint (Amir, 14 Apr 2026).

The practical distinction is decisive. $$s_{\alpha}=\int_{\mathbb{R}^2} x_1^{\alpha_1}x_2^{\alpha_2}\,d\mu(x_1,x_2),\qquad \alpha=(\alpha_1,\alpha_2)\in\mathbb{N}^2.$$8 uses all products $$s_{\alpha}=\int_{\mathbb{R}^2} x_1^{\alpha_1}x_2^{\alpha_2}\,d\mu(x_1,x_2),\qquad \alpha=(\alpha_1,\alpha_2)\in\mathbb{N}^2.$$9 and therefore has exponentially many generators in $K$0, while $K$1 uses only linear combinations of the $K$2 with SOS weights. Under Archimedean $K$3, semidefinite relaxations built from $K$4 are typically cheaper and converge in Lasserre’s hierarchy; Schmüdgen’s route guarantees convergence under mere compactness at the cost of larger cones (Amir, 14 Apr 2026).

The standard two-variable examples are explicit. On the unit disk,

$K$5

$K$6 is Archimedean with $K$7, so every polynomial strictly positive on the disk has a Putinar certificate

$K$8

On the box $K$9, the generators $\operatorname{supp}\mu\subseteq K$0, $\operatorname{supp}\mu\subseteq K$1 satisfy

$\operatorname{supp}\mu\subseteq K$2

so again $\operatorname{supp}\mu\subseteq K$3 is Archimedean (Amir, 14 Apr 2026).

3. Operator-theoretic constructions and analytic parametrizations

A complementary route starts from the sesquilinear form induced by the moments. Given $\operatorname{supp}\mu\subseteq K$4, define

$\operatorname{supp}\mu\subseteq K$5

The quotient $\operatorname{supp}\mu\subseteq K$6 becomes a pre-Hilbert space, whose completion $\operatorname{supp}\mu\subseteq K$7 carries densely defined symmetric multiplication operators

$\operatorname{supp}\mu\subseteq K$8

Under compactness or Archimedean hypotheses ensuring boundedness, these extend to bounded self-adjoint commuting operators, and the joint spectral theorem yields a measure $\operatorname{supp}\mu\subseteq K$9 on $s=(s_\alpha)_{\alpha\in\mathbb{N}^2}$0 with

$s=(s_\alpha)_{\alpha\in\mathbb{N}^2}$1

where $s=(s_\alpha)_{\alpha\in\mathbb{N}^2}$2. If $s=(s_\alpha)_{\alpha\in\mathbb{N}^2}$3 is nonnegative on $s=(s_\alpha)_{\alpha\in\mathbb{N}^2}$4, or on $s=(s_\alpha)_{\alpha\in\mathbb{N}^2}$5 under Putinar’s hypothesis, then $s=(s_\alpha)_{\alpha\in\mathbb{N}^2}$6 (Amir, 14 Apr 2026).

A more elaborate operator model solves the unconstrained two-dimensional Hamburger problem through an extended moment problem indexed by

$s=(s_\alpha)_{\alpha\in\mathbb{N}^2}$7

and moments

$s=(s_\alpha)_{\alpha\in\mathbb{N}^2}$8

The extended problem has a solution if and only if the positivity condition (P-Ext) and the difference relations (9)–(12), equivalently (13)–(16), hold; in that case the solution is unique. The original two-dimensional moment problem is solvable if and only if the given $s=(s_\alpha)_{\alpha\in\mathbb{N}^2}$9 can be embedded as $L:\mathbb{R}[x_1,x_2]\to\mathbb{R}$0 in such an extended sequence, and all solutions are then obtained by the paper’s iterative Hilbert-space algorithm (Zagorodnyuk, 2010). A second exposition of the same framework gives the same criterion and extends it to the complex moment problem through explicit conversion formulas between $L:\mathbb{R}[x_1,x_2]\to\mathbb{R}$1 and the real moments $L:\mathbb{R}[x_1,x_2]\to\mathbb{R}$2 (Zagorodnyuk, 2010).

Support restrictions can be incorporated operator-theoretically. For the strip

$L:\mathbb{R}[x_1,x_2]\to\mathbb{R}$3

solvability is equivalent to the positivity of the kernel

$L:\mathbb{R}[x_1,x_2]\to\mathbb{R}$4

together with the boundedness inequality

$L:\mathbb{R}[x_1,x_2]\to\mathbb{R}$5

for all finitely supported coefficients. All solutions are parameterized by spectral measures of one multiplication operator commuting with the spectral measure of the other, via generalized resolvents and decomposable contractions (Zagorodnyuk, 2010).

Under the Carleman-type condition

$L:\mathbb{R}[x_1,x_2]\to\mathbb{R}$6

all solutions of the two-dimensional moment problem are parameterized by analytic contractive operator-valued functions

$L:\mathbb{R}[x_1,x_2]\to\mathbb{R}$7

satisfying an intertwining constraint with the Cayley transform $L:\mathbb{R}[x_1,x_2]\to\mathbb{R}$8 of the second multiplication operator. In this setting, the determinate case is characterized by self-adjointness of the first multiplication operator $L:\mathbb{R}[x_1,x_2]\to\mathbb{R}$9, equivalently by the existence of a unique canonical solution (Zagorodnyuk, 14 Aug 2025).

4. Truncated problems, flat extensions, and canonical solutions

For truncated data $$L(p)=\sum_{\alpha} p_\alpha s_\alpha=\int_{\mathbb{R}^2} p(x_1,x_2)\,d\mu(x_1,x_2),$$0 with $$L(p)=\sum_{\alpha} p_\alpha s_\alpha=\int_{\mathbb{R}^2} p(x_1,x_2)\,d\mu(x_1,x_2),$$1, the moment matrix is

$$L(p)=\sum_{\alpha} p_\alpha s_\alpha=\int_{\mathbb{R}^2} p(x_1,x_2)\,d\mu(x_1,x_2),$$2

If $$L(p)=\sum_{\alpha} p_\alpha s_\alpha=\int_{\mathbb{R}^2} p(x_1,x_2)\,d\mu(x_1,x_2),$$3, each constraint polynomial $$L(p)=\sum_{\alpha} p_\alpha s_\alpha=\int_{\mathbb{R}^2} p(x_1,x_2)\,d\mu(x_1,x_2),$$4 produces a localizing matrix

$$L(p)=\sum_{\alpha} p_\alpha s_\alpha=\int_{\mathbb{R}^2} p(x_1,x_2)\,d\mu(x_1,x_2),$$5

The PSD conditions

$$L(p)=\sum_{\alpha} p_\alpha s_\alpha=\int_{\mathbb{R}^2} p(x_1,x_2)\,d\mu(x_1,x_2),$$6

are necessary for a $$L(p)=\sum_{\alpha} p_\alpha s_\alpha=\int_{\mathbb{R}^2} p(x_1,x_2)\,d\mu(x_1,x_2),$$7-supported representing measure, and under compact or Archimedean hypotheses they become asymptotically sufficient in the moment-SOS hierarchy (Amir, 14 Apr 2026).

The decisive finite-atomic criterion is flatness: $$L(p)=\sum_{\alpha} p_\alpha s_\alpha=\int_{\mathbb{R}^2} p(x_1,x_2)\,d\mu(x_1,x_2),$$8 If this holds together with $$L(p)=\sum_{\alpha} p_\alpha s_\alpha=\int_{\mathbb{R}^2} p(x_1,x_2)\,d\mu(x_1,x_2),$$9, then $\mathbb{R}^2$00 admits a finitely atomic representing measure

$\mathbb{R}^2$01

The kernel ideal $\mathbb{R}^2$02 determines algebraic relations satisfied by the atoms, and multiplication matrices on the quotient recover the atom locations through joint eigenvalues; the weights follow from a Vandermonde or evaluation system (Amir, 14 Apr 2026).

For triangular truncations

$\mathbb{R}^2$03

canonical solutions are defined as those generated by commuting self-adjoint extensions inside the associated Hilbert space. There is a one-to-one correspondence between canonical solutions and dimensionally stable close extensions, and in the two-dimensional triangular case this is a one-to-one correspondence between canonical solutions and flat extensions of the given data. The paper introduces the index of nonself-adjointness

$\mathbb{R}^2$04

with $\mathbb{R}^2$05 corresponding to flatness. For $\mathbb{R}^2$06, explicit necessary and sufficient conditions are obtained; for $\mathbb{R}^2$07, the existence question is reduced either to explicit conditions or to a single quadratic equation with several unknowns (Zagorodnyuk, 2024).

Rectangular truncations admit different constructive criteria. For the truncated two-dimensional problem with data $\mathbb{R}^2$08, one reduction places the representing measure on a finite union of parallel lines

$\mathbb{R}^2$09

and reduces solvability to $\mathbb{R}^2$10 one-dimensional truncated Hamburger problems via a Vandermonde system. This yields explicit necessary and sufficient conditions for $\mathbb{R}^2$11 and $\mathbb{R}^2$12, and explicit sufficient conditions for $\mathbb{R}^2$13, $\mathbb{R}^2$14, $\mathbb{R}^2$15, and $\mathbb{R}^2$16 (Zagorodnyuk, 2017).

5. Determinacy, algebraic varieties, and special support geometries

Determinacy means uniqueness of the representing measure. For compact $\mathbb{R}^2$17, determinacy is automatic. On $\mathbb{R}^2$18, one sufficient condition is the two-coordinate Carleman divergence

$\mathbb{R}^2$19

More generally, if the one-dimensional marginals are determinate, then the measure is determinate; this is Petersen’s principle. Operator-theoretic quasi-analyticity and essential self-adjointness criteria also imply determinacy (Amir, 14 Apr 2026).

For the Stieltjes problem on $\mathbb{R}^2$20, positivity

$\mathbb{R}^2$21

together with the multivariate Stieltjes divergence

$\mathbb{R}^2$22

implies existence and uniqueness of a representing measure on $\mathbb{R}^2$23. The same paper extends this to unbounded semialgebraic sets $\mathbb{R}^2$24, assuming positivity on $\mathbb{R}^2$25 and $\mathbb{R}^2$26 plus divergence along the generators $\mathbb{R}^2$27 (Schmüdgen, 2020).

Algebraic varieties provide a particularly transparent support condition. If $\mathbb{R}^2$28, then one can encode $\mathbb{R}^2$29 as the basic closed set $\mathbb{R}^2$30, and

$\mathbb{R}^2$31

Consequently, if $\mathbb{R}^2$32 on $\mathbb{R}^2$33, then for every $\mathbb{R}^2$34 there exist $\mathbb{R}^2$35 and $\mathbb{R}^2$36 such that $\mathbb{R}^2$37. In the moment problem on $\mathbb{R}^2$38, positivity on the variety is thus captured algebraically up to an arbitrarily small perturbation (Amir, 14 Apr 2026).

A substantial extension of this principle is the class of “curves with bumps.” If $\mathbb{R}^2$39 is a real principal ideal whose zero set $\mathbb{R}^2$40 satisfies the Scheiderer–Plaumann conditions, and $\mathbb{R}^2$41 is an Archimedean quadratic module, then

$\mathbb{R}^2$42

has the strong moment property, and its positivity set is

$\mathbb{R}^2$43

This yields non-negativity certificates and truncated SDP conditions for measures supported on a curve together with compact semialgebraic “bumps” (Kimsey et al., 2019).

Certain plane curves admit exact reduction to one-dimensional strong Hamburger problems. For the variety $\mathbb{R}^2$44, the two-dimensional truncated problem reduces to a strong truncated Hamburger problem with Laurent moments, and existence of a representing measure is equivalent to existence of a flat extension of the moment matrix for all degrees $\mathbb{R}^2$45. For $\mathbb{R}^2$46, the analogous reduction leads to a strong truncated Hamburger problem with one missing moment, which is solved through partial PSD conditions and Schur-complement formulas (Zalar, 2021).

6. Convex geometry, atom counts, and computational frameworks

In finite-dimensional truncated settings, the set of moment functionals forms a convex cone. For a finite-dimensional real vector space $\mathbb{R}^2$47, the moment cone is

$\mathbb{R}^2$48

Richter–Tchakaloff implies that every moment functional has an atomic representing measure with at most $\mathbb{R}^2$49 atoms (Dio et al., 2017). The moment cone is convex, and for differentiable $\mathbb{R}^2$50 one can analyze faces, exposed faces, inner points, and singularities through the differential of the atomic moment map

$\mathbb{R}^2$51

and through the sets $\mathbb{R}^2$52, $\mathbb{R}^2$53, and the core variety $\mathbb{R}^2$54, with

$\mathbb{R}^2$55

for nonzero $\mathbb{R}^2$56 (Dio et al., 2017).

For $\mathbb{R}^2$57, the Carathéodory number $\mathbb{R}^2$58 measures the maximal minimal number of atoms needed to represent an $\mathbb{R}^2$59-moment sequence. If

$\mathbb{R}^2$60

then the differential-geometric lower bound is

$\mathbb{R}^2$61

except for $\mathbb{R}^2$62, where $\mathbb{R}^2$63, and $\mathbb{R}^2$64, where $\mathbb{R}^2$65. On connected $\mathbb{R}^2$66, one has $\mathbb{R}^2$67 (Dio et al., 2017, Dio et al., 2018).

Several computational frameworks coexist with moment-SOS methods. In one analytic approach, the two-variable Cauchy or Stieltjes transform

$\mathbb{R}^2$68

is expanded at infinity, and a two-variable Schur algorithm produces continued-fraction representations for all solutions of certain truncated problems. In the non-symmetric and symmetric settings, every solution is described by finitely many Schur atoms together with a free remainder $\mathbb{R}^2$69; when $\mathbb{R}^2$70, the fraction terminates and the measure is atomic (Kovalyov et al., 2024).

A different computational viewpoint is maximum entropy on bounded domains. On $\mathbb{R}^2$71, the moment-constrained entropy maximizer has density

$\mathbb{R}^2$72

and the unknown multipliers solve nonlinear moment-matching equations. The equation-by-equation method solves these constraints sequentially through a homotopy/Newton scheme, can discard infeasible constraints, and in reported two-dimensional tests was more accurate than classical Newton’s method, MATLAB’s generic solver, and a BFGS-based method (Hao et al., 2017).

Taken together, these viewpoints show that the two-dimensional moment problem is not a single theorem but a family of closely related existence, uniqueness, extension, and reconstruction problems. Haviland-type positivity gives the abstract criterion; quadratic modules and preorderings supply semialgebraic certificates; GNS and spectral theory reconstruct measures from functionals; flat extensions and multiplication matrices recover atoms from truncated data; and convex, analytic, and optimization frameworks describe the geometry of solutions and the algorithms used to compute them (Amir, 14 Apr 2026).