RUI: Truncated Multivariable Moment Problems and Application: An Operator Theorectic Approach

摘要建议：DMS-9800805首席研究员：劳伦斯·菲尔科夫教授将研究与多变量矩问题的算子论方法有关的几个重要问题。这项研究的主要焦点是基于相关矩矩阵的扩张理论的多维截断矩问题的方法。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。表示度量的存在定理也可以解释为多变量加权移位的次正态完备性准则。这项研究的另一个方面涉及多维K矩问题，其中要求表示度量的支集包含在指定的闭集K中。这项研究的直接应用涉及数值分析中的多维求积问题。通过在求积的背景下应用矩阵扩展技术，本研究试图为圆盘、正方形、三角形或环形等集合构造最小求积规则。我们开发的算法可以明确地实现为计算机程序。数学矩被用来模拟许多真实世界的现象，如面积、速度、动量、加权平均、概率分布等。在使用积分计算矩时，最大的计算成本与函数调用有关。长期以来，科学家、数学家和工程师一直对设计有效的计算积分的方法感兴趣，这些方法可以最大限度地减少昂贵的函数调用数量。对于计算为直线上的积分的经典矩，这个问题可以用经典的数值分析工具，特别是高斯求积来解决。在目前的研究中，我们试图为平面上的集合(如三角形、圆盘或正方形)上的积分设计高效的计算算法。我们使用一种新的矩阵扩张方法来开发具体的计算规则，以有效地计算平面集合上的二维矩。这些规则可以被编码到计算机程序中，从而增强科学家和工程师有效地计算积分的能力。

Fialkow.Abs Abstract Proposal: Dms-9800805 Principal Investigator: Lawrence Fialkow Professor Fialkow will study several significant problems concerning an operator-theoretic approach to multivariable moment problems. The principal focus of this research is an approach to multidimensional truncated moment problems based on an extension theory for the associated moment matrix. When this matrix admits an infinite, positive, finite rank moment matrix extension, this method 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. Existence theorems for representing measures can also be interpreted as subnormal completion criteria for multivariable weighted shifts. Another facet of this research concerns the multidimensional K-moment problem, where the support of a representing measure is required to be contained in a prescribed closed set K. A direct application of this study concerns the Multidimensional Quadrature Problem in Numerical Analysis. By applying the matrix extension technique in the context of quadrature, this research seeks to construct minimal quadrature rules for such sets as the disk, square, triangle, or annulus. The algorithms that we develop can be explicitly implemented as computer programs. Mathematical moments are used to model many real-word phenomena, such as area, velocity, momentum, weighted averages, probability distributions, etc. In the evaluation of moments using integrals, the greatest computational costs are associated with function calls. It has long been of interest to scientists, mathematicians, and engineers to devise efficient computational methods for evaluating integrals, methods which minimize the number of expensive function calls. For classical moments, computed as integrals over a straight line, this problem is solved by classical tools of Numerical Analysis, particularly Gaussian Quadrature. In the present study we seek to devise highly efficient computational algorithms for integrals over sets in the plane (such as a triangle, disk, or square.) We use a new matrix extension method to develop concrete computational rules for efficient evaluation of two-dimensional moments over planar sets. These rules can be coded into computer programs which enhance the ability of scientists and engineers to evaluate integrals efficiently.