Truncated Multidimensional Trigonometric Moment Problem

• TMTMP is a central problem in harmonic analysis that constructs measures on a multi-dimensional torus to match prescribed trigonometric moments.
• Recent advances provide convex formulations and explicit basis representations that yield unique rational spectral density solutions for ARMA modeling.
• The method offers statistically optimal estimators and matrix-valued parametrizations, enhancing practical tractability in multivariate system identification.

The Truncated Multidimensional Trigonometric Moment Problem (TMTMP) is a central problem in harmonic analysis, mathematics of moments, and multivariate system identification. It concerns the characterization and construction of measures or functions on the $d$-dimensional torus whose finite collection of trigonometric moments matches prescribed data. TMTMP has particular significance in systems and signal processing, where one seeks solutions that also encode spectral properties, admit rational representations, and yield tractable estimation methods for modeling and control. Recent developments have produced new convex formulations, explicit basis choices, uniqueness guarantees, and efficient statistical estimation schemes, notably in the work of Wu and Lindquist (Wu et al., 13 Jan 2026). Additionally, there exist general matrix-valued and Nevanlinna-type parametrization results (Zagorodnyuk, 2012).

1. Formal Statement of the TMTMP

Let $d\geq 1$ denote the dimension, and let $\Omega \subset \mathbb{Z}^d$ be a finite, symmetric set containing the origin ($0\in\Omega$, $-\Omega=\Omega$). Given a real or complex sequence $\{c_{\boldsymbol k}\}_{\boldsymbol k\in\Omega}$ obeying $c_{-\boldsymbol k} = \overline{c_{\boldsymbol k}}$, the TMTMP asks for the existence (and, where possible, construction) of a nonnegative, bounded Radon measure $d\mu$ on the torus $\mathbb{T}^d = (-\pi,\pi]^d$ such that

$$c_{\boldsymbol k} = \int_{\mathbb{T}^d} e^{i\langle \boldsymbol k, \boldsymbol \theta\rangle}\,d\mu(\boldsymbol \theta),\quad \forall \boldsymbol k\in\Omega,$$

where $\langle \cdot,\cdot\rangle$ denotes the standard inner product. By Lebesgue’s decomposition, $d\mu$ may include absolutely continuous ($\Phi\,dm$) and singular components ($d\breve\mu$).

In system and signal processing, the interest lies in solutions $\Phi$ that are rational and strictly positive on $\mathbb{T}^d$, i.e., $\Phi = P/Q$ with $P,Q$ positive trigonometric polynomials, so that $\Phi$ can serve as a spectral density for an ARMA modeling problem—a formulation known as the multidimensional Rational Covariance Extension Problem (RCEP) (Wu et al., 13 Jan 2026).

2. Basis Function Selection and Polynomial Representation

Efficient treatment of the TMTMP depends crucially on the choice of basis for representing $Q$ and evaluating positivity. Wu and Lindquist (Wu et al., 13 Jan 2026) specialize to the hypercube index set of order $n$, i.e., $$\Omega = \{\boldsymbol k\in\mathbb{Z}^d: |k_j|\leq n,\;j=1,\ldots,d\}$$. The one-dimensional monomial vector is

$$K(e^{i\theta_j}) = [1, e^{i\theta_j},\ldots, e^{in\theta_j}]^T,$$

and the $d$-fold Kronecker product

$$K(e^{i\boldsymbol\theta}) = K(e^{i\theta_1}) \otimes \cdots \otimes K(e^{i\theta_d}) \in \mathbb{C}^{(n+1)^d}$$

enumerates all $$e^{i\langle \boldsymbol k,\boldsymbol \theta\rangle}$$, $\boldsymbol k\in\Omega$. Any positive trigonometric polynomial $Q$ can then be written as

$$Q(e^{i\boldsymbol \theta}) = K(e^{i\boldsymbol \theta})^H \Lambda K(e^{i\boldsymbol \theta}),$$

with $\Lambda \in \mathbb{C}^{(n+1)^d \times (n+1)^d}$ Hermitian and positive definite. This representation admits an explicit convex feasible domain for $\Lambda$: $$\mathfrak{L}^d_{+} = \{\,\Lambda: \Lambda = \Gamma_d(\Phi)\text{ for some }\Phi>0,\;\Lambda \succ 0\},$$ where $$\Gamma_d(\Phi) = \int_{\mathbb{T}^d} K \Phi K^H\,dm$$. This convex set is key to tractability and uniqueness.

3. Convex Optimization Formulation and Duality

The TMTMP admits a convex optimization formulation. For a prescribed reference $P(e^{i\boldsymbol\theta})>0$, consider the infinite-dimensional problem: $$\min_{\Phi \in C^0_+(\mathbb{T}^d)} \int_{\mathbb{T}^d} P\log\frac{P}{\Phi}\,dm,$$ subject to the moment constraints

$$\int_{\mathbb{T}^d} e^{i\langle \boldsymbol k, \theta\rangle} \Phi\,dm = c_{\boldsymbol k}, \;\forall\, \boldsymbol k\in\Omega.$$

The dual problem, invoking Lagrange multipliers $\Lambda$, becomes: $$\min_{\Lambda\in\mathfrak{L}^d_+}\;\; \mathbb{J}_P(\Lambda) = -\int_{\mathbb{T}^d} P\log(K^H\Lambda K)\,dm + \operatorname{tr}(\Lambda \mathcal{T}_d),$$ where $\mathcal{T}_d$ is the (block) Toeplitz matrix of moments. The solution $\Lambda^*\in\mathfrak{L}^d_+$ yields a unique rational $\Phi^* = P/(K^H\Lambda^* K)$ strictly positive on $\mathbb{T}^d$ (Wu et al., 13 Jan 2026).

4. Moment–Parameter Map, Existence, and Uniqueness

The mapping

$$\omega: \mathfrak{L}^d_+ \to \mathfrak{S}^d_+, \quad \omega(\Lambda) = \int_{\mathbb{T}^d} K \frac{P}{K^H \Lambda K} K^H\,dm$$

maps the cone $\mathfrak{L}^d_+$ onto the set $\mathfrak{S}^d_+$ of admissible positive-definite Toeplitz moment matrices. Wu and Lindquist prove that $\omega$ is a real-analytic diffeomorphism (bijective, smooth, open, and proper), so for each admissible $\mathcal{T}_d\succ 0$, the convex dual admits a unique minimizer and the primal TMTMP has a unique strictly positive rational solution (Wu et al., 13 Jan 2026).

5. Connections to RCEP and ARMA Process Modeling

If $P,Q$ are squared moduli of trigonometric polynomials $b,a$ of equal order, then

$\Phi^*(e^{i\theta}) = \frac{|b|^2}{|a|^2}$

is the spectrum of a multidimensional causal ARMA filter: $y(t_1,\dots,t_d) = \frac{b(\shift)}{a(\shift)}u(t_1,\dots,t_d),$ with $u$ white noise. The TMTMP thus delivers ARMA parameterizations that match prescribed covariance lags exactly, linking the moment problem to practical system identification and realization from empirical data (Wu et al., 13 Jan 2026).

6. Statistical Estimation Properties

When $N^d$ samples of a stationary $d$-variate process are observed, the sample trigonometric moments (biassed or unbiased) can be used to form an empirical Toeplitz matrix. Under mild regularity (e.g., mixing or Gaussianity):

• Consistency: As $N\to\infty$ and $n\to\infty$, the estimator $\hat\Phi_n(e^{i\theta})\to\Phi(e^{i\theta})$ in total variation.
• Asymptotic Unbiasedness: Unbiased moments yield unbiased estimators; bias vanishes as $N\to\infty$ for biased estimates.
• Convergence Rate: Sample moment variances scale as $O(1/N^d)$. The estimation achieves the parametric $1/\sqrt{N}$ rate, which is optimal.
• Efficiency: Under Gaussianity, the estimator is asymptotically efficient, attaining the Cramér–Rao bound (Wu et al., 13 Jan 2026).

7. General Matrix-Valued TMTMP and Nevanlinna-Type Parametrization

For the matrix-valued TMTMP, given $N\times N$ moment matrices $S_0,\dots,S_d$, the problem seeks a nondecreasing $N\times N$-valued function $M(t)$ on $[0,2\pi]$ such that $S_n = \int_0^{2\pi} e^{int} dM(t)$ for $n=0,\dots,d$. Solvability is equivalent to the positivity of the block Toeplitz matrix $T_d = [S_{i-j}]_{i,j=0}^d$ (Andô's theorem). Determinacy is reflected in the defect of a shift operator constructed on the quasi-Hilbert space defined by the moments, and explicit criteria are available (Zagorodnyuk, 2012). In the indeterminate case (defect $\delta\ge1$), solutions are parametrized via a linear fractional (Nevanlinna-type) formula involving the prescribed moments and an arbitrary analytic contraction, with all coefficients given explicitly. This provides a full parameterization of all solutions to the matrix TMTMP.

8. Algorithmic Solution and Simulation

Wu and Lindquist provide an explicit algorithm for TMTMP-based spectral estimation:

1. Compute both biased and unbiased sample moment sequences.
2. Use unbiased moments if the Toeplitz matrix is positive definite; else, use biased.
3. Solve the convex dual problem via Newton or BFGS methods to recover $\Lambda^*$.
4. Recover the unique spectral estimate $\hat\Phi(e^{i\theta}) = P/(K^H\Lambda^* K)$.

Empirical simulations confirm that this convex approach produces smooth spectral surfaces and accurate ARMA parameter recovery, outperforming non-convex least-squares methods, which may yield inconsistent or distorted estimates (Wu et al., 13 Jan 2026).

9. Summary Table: TMTMP Solution Methods

Approach	Characterization	Uniqueness
Convex Dual (Rational)	Explicit basis, convex feasible set	Unique (strictly pos.)
Nevanlinna Parametrizion	Matrix-fractional, contraction param.	All/extremal solutions

The convex dual formulation is especially significant for system applications, providing uniqueness, statistical optimality, and computational tractability. The Nevanlinna-type formula offers a full parametrization in the matrix-valued, generally indeterminate case, capturing the classical breadth of solutions.