The Truncated Univariate Rational Moment Problem: Characterizing Representing Measures on Arbitrary Closed Sets

Core Concepts

This paper provides a complete solution to the truncated univariate rational moment problem (K-RTMP) on arbitrary closed sets K, addressing a gap in previous research by characterizing the existence of representing measures that vanish on specific sets, including real poles.

Abstract

Bibliographic Information:

Nailwal, R., & Zalar, A. (2024). The truncated univariate rational moment problem. arXiv preprint arXiv:2411.11480v1.

Research Objective:

This paper aims to solve the truncated univariate rational moment problem (K-RTMP) for an arbitrary closed set K, which involves characterizing the existence of a positive Borel measure representing a given linear functional on a space of rational functions. The research specifically addresses the issue of ensuring the measure vanishes on specific sets, including real poles, a detail overlooked in previous work.

Methodology:

The authors utilize techniques from real algebraic geometry, particularly the theory of positive polynomials and moment matrices. They employ the concept of the natural description of a closed set and leverage results on characterizing positive polynomials on such sets. The solution involves converting the K-RTMP into a standard truncated moment problem (TMP) and analyzing the properties of localizing Hankel matrices. The authors differentiate between nonsingular and singular cases, providing distinct characterizations for each.

Key Findings:

1. Nonsingular Case: For a strictly K-positive linear functional, the paper proves the existence of a finitely atomic representing measure that vanishes on a given countable closed set. The authors further provide upper bounds on the number of atoms in the representing measure, particularly for semialgebraic sets.

2. Singular Case: The paper establishes necessary and sufficient conditions for the existence of a representing measure for a K-positive linear functional that is not strictly positive. This includes conditions related to the column relations of specific localizing Hankel matrices and the zeros of certain polynomials.

3. Counterexample and Applications: The authors provide a counterexample demonstrating the incompleteness of a previous solution to the K-RTMP for compact sets. They also apply their results to derive solutions for the strong Hamburger TMP and the TMP on the unit circle.

Main Conclusions:

The paper offers a comprehensive solution to the K-RTMP for arbitrary closed sets, addressing the previously overlooked aspect of ensuring the representing measure vanishes on specific sets. This work significantly extends the understanding of the K-RTMP and provides valuable tools for analyzing moment problems in various settings.

Significance:

This research contributes significantly to the field of moment problems, a classical area with connections to various mathematical disciplines, including operator theory, probability, and optimization. The results have implications for applications in areas such as signal processing, systems theory, and numerical analysis.

Limitations and Future Research:

While the paper provides a complete solution for the univariate case, extending these results to the multivariate rational moment problem remains an open question for future research. Further investigation into the computational aspects of constructing representing measures based on the provided characterizations would also be beneficial.

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Key Insights Distilled From

The truncated univariate rational moment problem

by Rajk... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2411.11480.pdf

Deeper Inquiries

How can the results of this paper be extended to solve the truncated multivariate rational moment problem?

Extending the results of this paper to the truncated multivariate rational moment problem (K-RTMP) presents a significant challenge, and a direct generalization is not straightforward. Here's why and some potential avenues for exploration:
Challenges:

Complexity of Positive Polynomials: In the univariate case, characterizing positive polynomials on a closed set K is manageable due to well-established results like those involving sums of squares and the natural description of K. However, in the multivariate setting, determining positivity of polynomials becomes vastly more intricate. The relationship between positive polynomials and sums of squares is not as straightforward, and finding tractable descriptions of positive polynomials on semi-algebraic sets is a current area of research in real algebraic geometry.
Lack of Factorization: The univariate case heavily relies on polynomial factorization. For instance, identifying the support of the representing measure from the zeros of certain polynomials is crucial. In multiple variables, polynomial factorization is not guaranteed, making this approach problematic.
Structure of Moment Matrices: The structure of Hankel matrices and their localizations, which are instrumental in the univariate K-RTMP, becomes significantly more complex in higher dimensions. Finding analogous matrix structures and developing corresponding criteria for the existence of representing measures is a nontrivial task.
Potential Approaches:

Restricting the Problem: One strategy is to focus on specific types of sets K and rational functions for which the multivariate K-RTMP becomes more tractable. For example, considering:

Product Sets: If K is a product of closed intervals in R^n, some results might be extended using techniques from multivariate orthogonal polynomials.
Symmetric Sets and Polynomials: Exploiting symmetry might simplify the problem. If K and the rational functions involved possess certain symmetries, the analysis of moment matrices could be simplified.

Functional Analysis and Operator Theory: As the second question suggests, these tools could offer valuable alternative perspectives. Representing the rational functions as operators on suitable function spaces might lead to new criteria for the existence of representing measures.
Numerical and Approximate Solutions: In cases where finding explicit solutions is difficult, developing numerical algorithms to approximate representing measures could be beneficial. Techniques from semi-definite programming and optimization could be particularly relevant.

Could there be alternative approaches, perhaps using techniques from functional analysis or operator theory, to address the K-RTMP and potentially yield different insights?

Yes, techniques from functional analysis and operator theory could provide valuable alternative approaches to the K-RTMP, potentially offering different insights and overcoming some limitations of purely algebraic methods. Here are some possibilities:

Operator Theoretic Representation:

Multiplication Operators: Consider the Hilbert space L^2(µ) of square-integrable functions with respect to a measure µ on K. Each rational function R in R(2k) can be viewed as a multiplication operator M_R on L^2(µ), where (M_R f)(x) = R(x)f(x).
Moment Problem Reformulation: The K-RTMP can be rephrased as a question about the existence of a positive Borel measure µ such that the linear functional L on R(2k) can be represented as L(R) = <f, M_R g>, where <.,.> is the inner product in L^2(µ), and f and g are suitable elements in L^2(µ).
Operator Theory Tools: This operator-theoretic framework allows us to leverage powerful tools from operator theory, such as:

Spectral Theory: The support of the measure µ is related to the spectrum of the multiplication operators M_R.
Positivity and Moment Sequences: Conditions for the existence of representing measures can be translated into conditions on the positivity of certain operators or sequences of operators.

Riesz Representation Theorem and its Generalizations:

Duality: The Riesz Representation Theorem establishes a duality between linear functionals on spaces of continuous functions and regular Borel measures. Generalizations of this theorem to spaces of functions with singularities (as we have with rational functions) could be explored to characterize representing measures for the K-RTMP.

Moment Problems in Abstract Settings:

Generalizations: The K-RTMP can be seen as a particular instance of a more general moment problem on algebras of functions. Results and techniques from the study of moment problems in abstract algebraic settings (e.g., on *-algebras) might provide new insights.

What are the practical implications of characterizing representing measures in this way for applications in fields like optimization or control theory?

Characterizing representing measures for the K-RTMP has significant practical implications in various fields, including optimization and control theory. Here are some examples:
Optimization:

Polynomial Optimization: Many optimization problems involve minimizing or maximizing polynomial functions subject to polynomial constraints. The K-RTMP is directly relevant when these constraints define a semi-algebraic set K.

Global Optimization: Finding global optima of polynomials is generally NP-hard. However, the K-RTMP, by providing a measure-theoretic representation of the problem, allows for the use of techniques like:

Moment Relaxation Hierarchies: These hierarchies provide a sequence of increasingly accurate approximations to the original polynomial optimization problem, often converging to the global optimum.
Semi-Definite Programming: The moment relaxations can often be formulated as semi-definite programs, which are convex optimization problems that can be solved efficiently.

Optimal Control: In optimal control, the goal is to find a control input for a dynamical system that minimizes a certain cost function, often subject to constraints on the system's state.

Occupation Measures: The K-RTMP can be used to characterize occupation measures, which describe the probability distribution of the system's state over time under a given control policy. This allows for the reformulation of optimal control problems as infinite-dimensional optimization problems over occupation measures.
Moment-Based Methods: Similar to polynomial optimization, moment-based methods can be applied to approximate the optimal control problem using a hierarchy of semi-definite programs.

Control Theory:

System Identification: The K-RTMP can be used to identify unknown parameters of a dynamical system from noisy measurements of its output. By formulating the system identification problem as a moment problem, one can estimate the system's parameters by finding a representing measure that best fits the observed data.

Robust Control: In robust control, the goal is to design controllers that maintain stability and performance despite uncertainties in the system's model. The K-RTMP can be used to characterize the set of all possible system behaviors consistent with the uncertainty, allowing for the design of controllers that are robust to these variations.

Key Benefits:

Global Optimality: Moment-based methods derived from the K-RTMP often provide guarantees of global optimality, unlike local optimization techniques that can get stuck in suboptimal solutions.
Tractability: The use of semi-definite programming makes these methods computationally tractable for problems of moderate size.
Flexibility: The K-RTMP framework is flexible and can be adapted to handle a wide range of optimization and control problems involving polynomial or rational functions.