Solving the Two-Dimensional Moment Problem Using the Schur Algorithm

Core Concepts

The article presents a solution to the truncated two-dimensional moment problem using the Schur algorithm, which is based on the continued fraction expansion of the solution. The results are applicable to the two-dimensional moment problem for atomic measures.

Abstract

The article studies the truncated two-dimensional moment problem and provides a solution using the Schur algorithm.

Key highlights:

1. The truncated two-dimensional moment problem is formulated in terms of the Stieltjes transform and the associated function F.
2. The non-symmetric case of the truncated problem is solved in Section 2, where the solutions are described by the Schur step-by-step algorithm.
3. The symmetric form of the truncated problem is studied in Section 3, and the solutions are expressed in terms of continued fractions.
4. A Stieltjes-like case is analyzed in Section 4, where the solutions are found using Stieltjes-like fractions.
5. The truncated two-dimensional moment problem for atomic measures is discussed in Section 5.
6. The set of solutions for the full two-dimensional problem is described in Section 6.

The article provides a comprehensive analysis of the two-dimensional moment problem and its solutions using the Schur algorithm and continued fractions.

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Key Insights Distilled From

Two-dimensional moment problem and Schur algorithm

by Ivan Kovalyo... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.03342.pdf

Deeper Inquiries

How can the solutions to the two-dimensional moment problem be extended to higher-dimensional cases

In the context of the two-dimensional moment problem, the solutions can be extended to higher-dimensional cases by generalizing the concepts and techniques used in the two-dimensional case. This extension involves considering moments in multiple dimensions, such as three-dimensional or higher-dimensional spaces. The key idea is to define the moment sequences and associated functions in higher dimensions, similar to how it is done in the two-dimensional case. By adapting the Stieltjes transform, Schur algorithm, and continued fraction representations to higher dimensions, it is possible to formulate and solve moment problems in multidimensional spaces.

What are the potential applications of the two-dimensional moment problem solutions in areas such as signal processing, image analysis, or numerical analysis

The solutions to the two-dimensional moment problem have various potential applications in signal processing, image analysis, and numerical analysis. In signal processing, moment-based features extracted from signals can be used for pattern recognition, classification, and noise reduction. In image analysis, moments can capture shape, texture, and other characteristics of images, enabling tasks like object recognition and image enhancement. In numerical analysis, moment methods can be applied to solve integral equations, optimization problems, and probability distributions. The solutions obtained from the two-dimensional moment problem provide a foundation for developing algorithms and techniques that can be applied in these areas.

Can the Schur algorithm and continued fraction representations be used to study other types of multidimensional moment problems or related mathematical structures

Yes, the Schur algorithm and continued fraction representations can be used to study other types of multidimensional moment problems or related mathematical structures. By extending the principles of the Schur algorithm and continued fractions to higher dimensions, it is possible to analyze and solve moment problems in multiple dimensions. This approach can be applied to problems involving moments in spaces of higher dimensionality, such as three-dimensional or four-dimensional spaces. The techniques can also be adapted to study related mathematical structures, such as multivariate polynomials, orthogonal polynomials in multiple variables, and multidimensional interpolation problems. The versatility of the Schur algorithm and continued fraction representations makes them valuable tools for exploring a wide range of multidimensional mathematical problems.