Indeterminate Hamburger Moment Problem

Updated 23 November 2025

The indeterminate Hamburger moment problem is characterized by infinitely many positive measures sharing the same moment sequence, highlighting non-uniqueness.
It employs operator theory by linking moment sequences to Jacobi operators, where self-adjoint extensions and deficiency indices govern the parametrization.
Applications span spectral analysis and entropy maximization, with criteria based on Hankel eigenvalues and analytic invariants guiding the classification.

The indeterminate Hamburger moment problem is a central object in the theory of moment sequences, harmonic analysis, operator theory, and the spectral theory of unbounded symmetric operators. In this context, "indeterminate" refers to the existence of infinitely many positive Borel measures on ℝ that share the same sequence of moments, in contrast to the "determinate" case where uniqueness holds. The dichotomy between determinacy and indeterminacy is governed by deep analytic, algebraic, and operator-theoretic criteria, making the structure and parametrization of the indeterminate case a rich subject at the interface of several mathematical domains.

1. Definitions and Fundamental Criteria

Given a real sequence $(s_n)_{n\ge0}$ , the (classical) Hamburger moment problem asks for a positive Borel measure μ on $\mathbb R$ such that $s_k = \int_{\mathbb R} t^k\,d\mu(t)$ for all $k\ge0$ , usually normalized such that $s_0=1$ . If μ is unique, the problem is called determinate; if there are infinitely many solutions, it is indeterminate (Bustos et al., 2021, Inverardi et al., 2023).

Classical criteria for (in)determinacy are as follows:

Criterion	Determinacy	Indeterminacy
Carleman	$\sum_{n=1}^\infty m_{2n}^{-1/(2n)} = \infty$	(If sum converges, indeterminacy is possible)
Krein		$\int_{-\infty}^\infty \frac{\ln w(x)}{1+x^2}\,dx > -\infty$
Hankel Eigenvalues	$\lambda_{\min}(H_n) \to 0$	$\lambda_{\min}(H_n) \to c_0 > 0$

Here $m_{2n}$ denotes even moments, $w(x)$ an absolutely continuous part of μ, and $H_n$ the (n+1)×(n+1) Hankel matrix $[s_{i+j}]_{i,j=0}^n$ (Lin et al., 2014, Stoyanov et al., 2019, Inverardi et al., 2023).

2. Operator-Theoretic Structure and Parametrization

There is a canonical correspondence between normalized Hamburger moment sequences and semi-infinite Jacobi operators (real symmetric tridiagonal matrices): $J = \begin{pmatrix} q_1 & b_1 & 0 & \cdots \ b_1 & q_2 & b_2 & \ddots \ 0 & b_2 & q_3 & \ddots \ \vdots & \ddots & \ddots & \ddots \end{pmatrix}, \quad b_k > 0,\ q_k\in\mathbb R.$ The self-adjointness of $J$ is a spectral criterion for determinacy: $J$ essentially self-adjoint (i.e., has deficiency indices $(0,0)$ ) if and only if the moment problem is determinate, while deficiency indices $(1,1)$ correspond to indeterminacy. The non-selfadjoint case yields a one-parameter family of self-adjoint extensions, reflecting the nonuniqueness of representing measures (Bustos et al., 2021, Berg et al., 2023, Mikhaylov et al., 2019, Zagorodnyuk, 2016).

Indeterminate moment problems are parameterized via the Nevanlinna matrix $W(z)$ , with entries $A(z), B(z), C(z), D(z)$ , all entire functions of minimal exponential type and satisfying $A(z)D(z) - B(z)C(z) = 1$ . Every solution measure $\mu$ corresponds to a unique Herglotz function $T$ via

$\int_{\mathbb R} \frac{d\mu(t)}{t-z} = \frac{A(z)T(z)+B(z)}{C(z)T(z)+D(z)} \quad (z\in\mathbb C^+).$

Special cases include constant $T(z)$ (absolutely continuous solutions) and real $T$ corresponding to N-extremal measures (spectral measures of self-adjoint extensions). This parametrization extends to the operator-valued setting (Zagorodnyuk, 2016).

3. Algebraic and Analytic Criteria

Classical criteria for indeterminacy include:

Krein's criterion: If $\int_{-\infty}^{\infty} \frac{\log \mu'(x)}{1+x^2} \, dx > -\infty$ , then the problem is indeterminate (Lin, 2017, Stoyanov et al., 2019).
Converse to Carleman's criterion: If the Carleman sum converges and the density satisfies Lin's condition ( $-x f'(x)/f(x) \to \infty$ ), then the moment problem is indeterminate (Lin et al., 2014, Lin, 2017).
Smallest Hankel eigenvalue: $\lim_{n\to\infty}\lambda_{\min}(H_n) = c_0 > 0$ is both necessary and sufficient for indeterminacy (Inverardi et al., 2023, Berg et al., 2018).

Quantitative refinements use the order $\rho$ and type $\tau$ of the associated entire functions in the Nevanlinna matrix. For symmetric problems, the order is determined by the harmonic mean of asymptotic exponents of orthonormal polynomials: $\rho \leq \gamma = \frac{2}{1/\alpha + 1/\beta}$ for $P_{2n}(0)^2 \sim c_1 n^{-1/\beta}$ and $Q_{2n-1}(0)^2 \sim c_2 n^{-1/\alpha}$ , $0 < \alpha, \beta < 1$ (Berg et al., 2015, Pruckner et al., 2015, Pruckner et al., 2023).

4. Rigidity, Inverse, and Structural Properties

A crucial property of indeterminate Hamburger moment sequences is non-rigidity: any sufficiently small perturbation of finitely many moments yields another indeterminate moment sequence. Determinate sequences with infinite index are rigid (except for the zeroth moment); determinate sequences with finite index allow only finitely many deformations (Dyachenko, 2017).

In the indeterminate case, the infinite Hankel matrix $\mathcal H = \{s_{m+n}\}$ admits an infinite family of symmetric inverses under absolute convergence of the defining series—given, for example, by the reproducing kernel of the associated orthogonal polynomials. Strong sufficient conditions for such invertibility include rapid decay of recurrence coefficient ratios $b_{n-1}/b_n < q < 1$ or suitable subexponential decay. The inverse is highly non-unique, reflecting the underlying indeterminacy (Berg et al., 2018).

From a functional analytic perspective, the Hankel quadratic form and associated operator are always closable in the indeterminate case and, under certain circumstances, for determinate sequences of finite index. Closability is thus not an indicator of uniqueness; only strong moment decay can force support restrictions (Berg et al., 2019).

5. Examples, Applications, and Quantitative Invariants

Canonical examples of indeterminate Hamburger moment sequences include:

Lognormal distributions: $m_k = \exp(\sigma^2 k^2/2)$ is indeterminate—both Carleman's and Krein's criteria diagnose indeterminacy (Lin et al., 2014, Lin, 2017, Berg et al., 2015).
Stieltjes–Wigert and certain $q$ -Hermite polynomial systems: the asymptotic behavior of recurrence coefficients ensures indeterminacy (Berg et al., 2018).
Powers and products of classical distributions (e.g., Normal, Exponential): suitable growth rate conditions for moments yield indeterminacy (Lin et al., 2014, Lin, 2017).

The behavior of the Nevanlinna matrix determines entropy and maximum entropy solutions. Every indeterminate Hamburger moment sequence supports a family of analytic densities with finite Shannon entropy, guaranteeing well-posedness and attainment of maximum entropy within the convex set of solutions (Berg, 16 Nov 2025).

In the operator-theoretic setting, the parametrization of all self-adjoint extensions of the Jacobi operator, and hence of all representing measures, is explicit and computable via the properties of the associated canonical system (Berg et al., 2023, Pruckner et al., 2015, Pruckner et al., 2023). The spectral measures of self-adjoint extensions correspond to extremal or discrete solutions in the Nevanlinna family.

6. Basis Multiplicity and New Operator-Algebraic Characterizations

A novel criterion for the determinate–indeterminate dichotomy, due to Hernández Bustos–Palafox–Silva, uses the Akhiezer–Glazman representation theory for Jacobi operators: there is more than one orthonormal basis of representation producing a tridiagonal Jacobi matrix if and only if the operator is self-adjoint (i.e., determinate); if there is only one such basis (up to sign), the operator is symmetric non-self-adjoint (i.e., indeterminate). This operator-theoretic/algebraic approach complements classical growth and spectral criteria and suggests algorithmic methods for testing indeterminacy via searching for alternative Jacobi representations (Bustos et al., 2021).

7. Extensions, Further Directions, and Open Problems

The structural theory extends to strong Hamburger moment problems (moments indexed by $\mathbb Z$ , not just $\mathbb N$ ), where Aldén generalized classical results, proving that Krein’s integrability condition suffices for indeterminacy even without symmetry. Explicit construction of entire densities, analysis of null-spaces, and more refined bounds on Nevanlinna invariants (order, type) continue to advance understanding (Aldén, 2016, Berg et al., 2015).

Generalizations to multidimensional and operator-valued moment problems, matrix moment problems, and Stieltjes/Hausdorff analogues remain active research areas. Sharp computable criteria for the order and type of indeterminate Nevanlinna matrices, determination of maximum entropy solutions, and structural description of the convex set of solutions are all significant directions (Pruckner et al., 2015, Berg, 16 Nov 2025, Zagorodnyuk, 2016, Pruckner et al., 2023).

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