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

RUI：截断多变量矩问题及应用：算子理论方法

• 批准号: 9800805
• 负责人: Lawrence Fialkow
• 金额: $ 5.74万
• 依托单位: SUNY College at New Paltz
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800805&HistoricalAwards=false
• 关键词: RUI; Truncated; Multivariable; Moment; Problems

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.

Fialkow.Abs 摘要 提案：DMS-9800805主要研究者：Lawrence Fialkow Fialkow教授将研究几个重要的问题，关于运营商的理论方法，多变量的时刻问题。本研究的主要重点是一种方法，以多维截断矩问题的扩展理论的基础上相关联的时刻矩阵。当这个矩阵允许一个无限的，正的，有限秩矩量矩阵的扩展，这种方法产生一个显式的公式，支持一个正常元组的运营商相应的扩展的联合频谱上的原子表示措施。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。表示措施的存在定理也可以解释为多变量加权移位的次正常完成准则。本研究的另一个方面涉及多维K-矩问题，其中表示测度的支持度需要包含在指定的闭集K中。本研究的一个直接应用涉及数值分析中的多维求积问题。通过应用矩阵扩展技术的背景下，求积，本研究旨在构建最小的求积规则，如磁盘，正方形，三角形，或环形。我们开发的算法可以明确地实现为计算机程序。 数学矩用于模拟许多真实世界的现象，如面积，速度，动量，加权平均值，概率分布等，在使用积分的矩的评估，最大的计算成本与函数调用。科学家、数学家和工程师一直对设计有效的计算方法来计算积分感兴趣，这些方法可以最大限度地减少昂贵的函数调用次数。对于经典矩，计算为直线上的积分，这个问题可以通过经典的数值分析工具来解决，特别是高斯求积。在本研究中，我们试图设计高效的计算算法的积分集在平面上（如三角形，磁盘或正方形）。我们使用一个新的矩阵扩展方法来开发具体的计算规则，有效地评估二维矩平面集。这些规则可以编码成计算机程序，提高科学家和工程师的能力，有效地评估积分。