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

PI：Lawrence A.Fialkow，SUNY New PaltzDMS-0201430摘要本研究基于相关矩矩阵的扩张理论，研究了多维截断功率矩问题。当该矩阵允许无限、正、有限秩矩矩阵扩张时，该方法得到了与该扩张对应的正规算子组的联合谱上所支持的有限原子表示测度的显式公式。这项研究的目的是确定允许所需扩展的时刻数据的具体条件。这方面的研究主要涉及多变量截断K-矩问题，其中表示度量的支持度必须包含在指定的闭集K中。对于K-代数簇，本研究提出了一种新的猜想，用于用与矩数据密切相关的具体代数和几何不变量来求解矩问题。对于K-半代数，本研究试图将Curto-Fialkow提出的截断K-矩问题的抽象解应用于特定的半代数集，如闭单位圆盘。这项研究的一个直接应用涉及数值分析中的多维体积问题。通过将矩矩阵扩展技术应用于体积，本研究试图为圆盘、正方形、三角形或环形等集合构造最小体积规则。本研究的一个方面，求积问题，涉及不规则区域大小的有效测量，或密度不均匀分布的体积的重量的测量。我们试图识别身体内的少量测试点(或节点)，通过测量这几个点的密度，我们可以接近身体的整体大小或重量。在线性物体的情况下，例如细杆，数学家卡尔·弗里德里希·高斯(1777-1855)用一种现在被称为高斯求积的技术，用最少的测试点解决了求积问题。目前，令人惊讶的是，对于平面或三维空间中的形状，甚至对于基本集合，如圆盘或球体，已知的最小节点求积规则很少。为了研究求积问题，我们实际上研究了一个更一般的问题，即多维截断矩问题。许多真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，这些属性与系统底层的物理空间有关。截断矩问题询问系统是否由它的矩序列唯一确定，以及系统的矩是否可以通过研究系统在底层空间中的有限个节点来计算。在这项研究中，我们开发了算法来识别何时存在这样的节点序列，并高效地计算它们。通过这种方式，我们可以仅用底层空间中有限数量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.