Truncated Multidimensional Trigonometric Moment Problem: A Choice of Bases and the Unique Solution

In this paper, we resolve the Truncated Multidimensional Trigonometric Moment Problem (TMTMP) from a system and signal processing perspective, which serves as the foundation for the Multidimensional Rational Covariance Extension Problem (RCEP).

While standard mathematical TMTMPs focus on the existence of atomic measure solutions, system identification requires analytic rational solutions with positive polynomial coefficients.

To overcome the long-standing challenge of characterizing the positive feasible domain under general bases, we propose a novel choice of basis functions and a corresponding estimation scheme via convex optimization.

We establish an explicit condition to guarantee the positiveness of the spectral estimate.

Crucially, the map from the estimate parameters to the trigonometric moments is proved to be a diffeomorphism, ensuring the existence and uniqueness of the solution.

Furthermore, we comprehensively prove the statistical properties of the estimator, including its consistency, (asymptotic) unbiasedness, convergence rate, and efficiency.

The proposed framework is applied to a system identification task, where simulations validate its effectiveness.