Moment problem
A moment problem asks whether a prescribed sequence of numbers can be represented as the moments of a nonnegative Borel measure, and, if such a measure exists, whether it is uniquely determined. For a sequence ((m_n)_{n\geq 0}), the basic representation has the form
[ m_n=\int_K x^n,d\mu(x),\qquad n=0,1,2,\ldots, ]
where (K\subseteq\mathbb R) is a specified support set and (\mu) is a nonnegative measure for which every displayed integral is finite. The location of (K) substantially changes both the existence criteria and the uniqueness theory.
The classical cases are the Hamburger moment problem, in which (K=\mathbb R); the Stieltjes moment problem, in which (K=[0,\infty)); and the Hausdorff moment problem, in which (K=[0,1]). These problems connect measure theory with positive linear functionals, Hankel matrices, orthogonal polynomials, and the spectral theory of Jacobi operators.
Historical development
The terminology originated in the study of integral representations and continued fractions during the late nineteenth century. Thomas Joannes Stieltjes formulated the half-line problem while investigating analytic continued fractions and functions represented by positive measures. His work identified moments as coefficients encoding a measure, while also demonstrating that such coefficients do not always determine that measure uniquely.
Hans Hamburger subsequently established a systematic existence theory for measures on the full real line. Felix Hausdorff treated bounded intervals and obtained a criterion expressed entirely through finite differences. Later formulations by Marcel Riesz placed the subject within the theory of positive linear functionals, thereby separating the algebraic data of the moments from the representing measure.
During the twentieth century, the theory became closely integrated with spectral analysis. Naum Akhiezer developed the analytic structure of indeterminate problems, while Jacob Shohat and John Tamarkin organized the classical results around orthogonal polynomials and integral representations.
Algebraic formulation
A sequence ((m_n)) defines a linear functional (L) on the polynomial ring (\mathbb R[x]) by
[ L!\left(\sum_{k=0}^{N}a_kx^k\right) =\sum_{k=0}^{N}a_km_k. ]
A representing measure supported on (K) satisfies
[ L(p)=\int_K p(x),d\mu(x). ]
Consequently, (L(p)\geq 0) whenever (p) is nonnegative on (K). The moment problem therefore becomes a question about which linear functionals are positive on the cone of polynomials that remain nonnegative over the prescribed support.
For the real-line problem, every nonnegative univariate polynomial is a sum of polynomial squares. Existence is therefore equivalent to
[ L(q^2)\geq 0 ]
for every polynomial (q). Writing (q(x)=\sum_{j=0}^{N}c_jx^j) gives
[ L(q^2)=\sum_{i,j=0}^{N}c_ic_jm_{i+j}. ]
Thus the infinite Hankel matrix
[ H=(m_{i+j})_{i,j\geq0} ]
must be positive semidefinite. Conversely, positivity of every finite principal section of (H) yields a representing measure on (\mathbb R).
For support on the nonnegative half-line, positivity must also hold after multiplication by (x). The Stieltjes existence criterion is therefore equivalent to positive semidefiniteness of both families
[ H_N=(m_{i+j})_{i,j=0}^{N} ]
and
[ H_N^{(1)}=(m_{i+j+1})_{i,j=0}^{N}. ]
The second family encodes the fact that (xq(x)^2) is nonnegative on ([0,\infty)).
The compact-interval case
For a measure supported on ([0,1]), the sequence is characterized by complete monotonicity under the forward-difference operator
[ \Delta m_n=m_{n+1}-m_n. ]
Hausdorff’s criterion states that ((m_n)) is a moment sequence on ([0,1]) exactly when
[ (-1)^k\Delta^k m_n\geq0 ]
for every pair of nonnegative integers (n) and (k). The identity underlying this condition is
[ (-1)^k\Delta^k m_n =\int_0^1 x^n(1-x)^k,d\mu(x), ]
whose right-hand side is nonnegative.
In 1936, You Watanabe expressed the same criterion for an arbitrary finite interval ([a,b]) through affine normalization. Under the coordinate transformation
[ y=\frac{x-a}{b-a}, ]
the normalized moments are
[ \widetilde m_n =\frac{1}{(b-a)^n} \sum_{j=0}^{n} \binom{n}{j}(-a)^{,n-j}m_j. ]
Watanabe showed that the representing measure is supported on ([a,b]) precisely when the transformed sequence satisfies the Hausdorff finite-difference inequalities. The transformation introduced no additional existence condition; it transferred the compact-interval problem to the canonical interval while preserving the total mass (\widetilde m_0=m_0).
Compact support also resolves the uniqueness question. If two finite measures on ([a,b]) have identical moments, then they agree on all polynomials. The Stone–Weierstrass theorem implies that polynomials are uniformly dense in the continuous functions on this interval, so the measures agree on every continuous test function and are therefore identical.
Determinate and indeterminate sequences
A moment sequence is called determinate when it has exactly one representing measure. It is indeterminate when distinct measures produce the same complete sequence of moments. Existence and determinacy are separate properties: positivity of the relevant Hankel forms establishes existence but does not generally establish uniqueness.
For the Hamburger problem, Carleman’s condition
[ \sum_{n=1}^{\infty}m_{2n}^{-1/(2n)}=\infty ]
is sufficient for determinacy. The condition controls the growth of the even moments and prevents the representing measure from varying while retaining every polynomial integral. It is not a necessary condition, so convergence of the series does not by itself imply indeterminacy.
The log-normal distribution provides a standard indeterminate Stieltjes sequence. Its moments grow rapidly enough that distinct measures on the positive half-line share the same values. Such examples demonstrate that knowledge of all integer moments does not universally determine a probability distribution, even when every moment is finite.
Determinacy also has an operator-theoretic interpretation. Orthogonalization of (1,x,x^2,\ldots) with respect to the moment functional produces a sequence of orthogonal polynomials satisfying a three-term recurrence. The recurrence coefficients define a symmetric Jacobi operator, and essential self-adjointness of that operator corresponds to uniqueness in the associated Hamburger problem. When the operator has nontrivial self-adjoint extensions, their spectral measures form a family of representing measures for the same sequence.
Orthogonal polynomials and continued fractions
Given a positive-definite moment functional, the Gram–Schmidt process applied to the monomials produces orthonormal polynomials ((p_n)). They satisfy a recurrence of the form
[ xp_n(x)=a_{n+1}p_{n+1}(x)+b_np_n(x)+a_np_{n-1}(x), ]
where (a_n>0) and (b_n\in\mathbb R). The corresponding tridiagonal matrix is the Jacobi matrix associated with the moment sequence.
The measure’s Cauchy transform,
[ F(z)=\int_{\mathbb R}\frac{d\mu(x)}{z-x}, ]
has the asymptotic expansion
[ F(z)\sim \frac{m_0}{z}+\frac{m_1}{z^2} +\frac{m_2}{z^3}+\cdots ]
away from the support. The recurrence coefficients generate a continued fraction for (F), linking the moment problem to classical analytic continued fractions. In an indeterminate problem, the formal asymptotic expansion remains fixed while the analytic function varies within a parameterized family.
Truncated moment problems
A truncated moment problem specifies only finitely many values,
[ m_0,m_1,\ldots,m_{2d}. ]
The resulting question concerns measures whose first (2d+1) moments match the data. Positive semidefiniteness of the finite moment matrix
[ M_d=(m_{i+j})_{i,j=0}^{d} ]
is necessary for a real-line representing measure. In the univariate setting it also forms the central algebraic existence condition, supplemented by localizing matrices when the support is restricted by polynomial inequalities.
Finite moment data generally admit many representing measures. Their collection is a convex set, while finitely supported measures occur at its extremal boundary. Rank conditions on moment matrices determine when a finite atomic representation exists and how many support points it requires. This finite-dimensional structure connects the truncated problem with convex geometry and semidefinite programming.