RUI: Truncated Multivariable Moment Problems & Applications: An Operator Theoretic Approach

摘要本文研究了一种基于关联矩矩阵可拓理论的多维截断功率矩问题的求解方法。当这个矩阵允许无限的、正的、有限秩矩矩阵扩展时，这种方法得到了一个有限原子表示测度的显式公式，该测度支持在与该扩展相对应的正常元组算子的联合谱上。本研究的目的是确定允许所需扩展的力矩数据的具体条件。该研究主要关注多变量截断K矩问题，其中要求一个表示测度的支持包含在一个规定的闭集K中。对于K是一个代数变量，本研究关注一个新的猜想，该猜想可以用与矩数据密切相关的具体代数和几何不变量来解决矩问题。对于K半代数，本研究试图将Curto-Fialkow截断K矩问题的抽象解应用于特定的半代数集，如封闭单元盘。本研究的一个直接应用涉及数值分析中的多维培养问题。通过将矩矩阵扩展技术应用于立方体环境中，本研究试图构建诸如圆盘、正方形、三角形或环空等集合的最小立方体规则。该研究的一个方面是正交问题，它涉及对不规则区域大小的有效测量，或者对密度分布不均匀的体积重量的测量。我们试图确定身体内的少量测试点（或节点），通过测量这些点的密度，我们可以接近身体的总体尺寸或重量。在线性物体的情况下，比如细杆，正交问题是由数学家卡尔·弗里德里希·高斯（1777-1855）用最少的测试点解决的，他使用了一种现在被称为高斯正交的技术。目前，对于平面或三维空间中的形状，甚至对于像圆盘或球体这样的基本集合，已知的最小节点正交规则少得惊人。为了研究正交问题，我们实际上研究了一个更一般的问题，多维截断矩问题。许多现实世界的系统可以用一系列称为矩的物理属性来描述，例如质量、重量、动量等，它们与系统底层的物理空间有关。截断矩问题问的是一个系统是否唯一地由它的矩序列决定，以及一个系统的矩是否可以通过研究系统在底层空间中有限数量的节点来计算。在这项研究中，我们开发了一种算法来识别这种节点序列的存在，并有效地计算它们。这样，我们就可以用底层空间中有限个数的点来描述一个系统。

PI: Lawrence A. Fialkow, SUNY New PaltzDMS-0201430 AbstractThis research concerns an approach to the multidimensional truncated power moment problem based on an extension theory for the associated moment matrix. When this matrix admits an infinite, positive, finite rank moment matrix extension, this approach yields an explicit formula for a finitely atomic representing measure supported on the joint spectrum of a normal tuple of operators corresponding to the extension. The aim of this research is to determine concrete conditions on the moment data which permit the desired extension. Much of this research concerns, specifically, the multivariable truncated K-moment problem, where the support of a representing measure is required to be contained in a prescribed closed set K. For K an algebraic variety, this research concerns a new conjecture for solving the moment problem in terms of concrete algebraic and geometric invariants closely related to the moment data. For K semi-algebraic, this research seeks to apply the abstract solution of the truncated K-moment problem due to Curto-Fialkow to specific semi-algebraic sets such as the closed unit disk. A direct application of this study concerns the multidimensional cubature problem in Numerical Analysis. By applying the moment matrix extension technique in the context of cubature, this research seeks to construct minimal cubature rules for sets such as the disk, square, triangle, or annulus.One aspect of this research, the Quadrature Problem, concerns the efficient measurement of the size of an irregular area, or the measurement of the weight of a volume whose density is unevenly distributed. We seek to identify a small number of test points (or nodes) within the body in such a way that by measuring the density just at these few points, we may closely approximate the overall size or weight of the body. In the case of a linear body, such as a thin rod, the quadrature problem was solved with the fewest number of test points by the mathematician Carl Friedrich Gauss (1777-1855), using a technique now known as Gaussian Quadrature. At the present time, surprisingly few minimal-node quadrature rules are known for shapes in the plane or in 3-dimensional space, even for basic sets such as a disk or sphere. In order to study the Quadrature Problem, we actually study a more general problem, the Multidimensional Truncated Moment Problem. Many real-world systems can be described by a sequence of physical attributes called moments, such as mass, weight, momentum, etc., which relate to the physical space underlying the system. The truncated moment problem asks whether a system is uniquely determined by its sequence of moments, and also whether the moments of a system can be computed by studying the system at just a finite number of nodes in the underlying space. In this research, we develop algorithms for recognizing when such a sequence of nodes exists, and for efficiently computing them. In this way, we may describe a system in terms of just a finite number of points in the underlying space.