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

RUI：截断多变量矩问题

• 批准号: 0758378
• 负责人: Lawrence Fialkow
• 金额: $ 17.5万
• 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: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0758378&HistoricalAwards=false
• 关键词: RUI; Truncated; Multivariable; Moment; Problems

Truncated Multivariable Moment Problems and Applications: An Operator Theoretic Approach This research concerns an operator-theoretic approach to multivariable truncated moment problems. For a finite real multi-sequence, we seek concrete necessary and sufficient conditions for the existence of a positive Borel representing measure in Euclidean space, a measure such that the power moments of the measure coincide with the corresponding elements of the multi-sequence. We associate to the multi-sequence a corresponding moment matrix. It is known that a representing measure exists if and only if the moment matrix admits an extension to a larger, positive semi-definite moment matrix, which in turn admits a flat, i.e., rank-preserving, extension to a still larger moment matrix. This research concerns the existence and minimal size of such moment matrix extensions. Column dependence relations in the moment matrix determine an algebraic variety which contains the support of any representing measure, and this research concerns a description of the algebraic varieties for which it is possible to establish the desired flat moment matrix extensions. More generally, this research concerns representing measure whose support is contained in a prescribed semi-algebraic closed set. In this case, we require flat extensions (as above) for which the localizing matrices corresponding to the semi-algebraic set are also positive semi-definite. This part of the research is concerned with semi-algebraic (or algebraic) sets for which positive polynomials admit degree-bounded weighted sum-of-squares representations; applications directly concern finite convergence in Lasserre's polynomial optimization theory. Another aspect of this research concerns algorithms for explicitly computing finitely atomic representing measures; applications of this part of the research lead to the construction of minimal or near-minimal multivariable cubature rules in Numerical Analysis. The aim of this research is to develop new existence and uniqueness criteria for finitely atomic representing measures in multivariable truncated moment problems. Truncated moment problems play an essential role in aspects of such fields as Operator Theory (subnormality of weighted shifts), Interpolation Theory (classical Nevanlinna-Pick theory), Numerical Analysis (multivariable cubature rules), Control Theory (signal processing), Optimization Theory (polynomial optimization over a region), and Real Algebraic Geometry (representations of positive polynomials as weighted sums-of-squares). The principal focus of this research is an approach to multidimensional truncated moment problems based on an extension theory for the moment matrix associated to the moment data. 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 primary goal of this research is to determine concrete conditions on the moment data which permit the desired extension. This research also concerns algorithms for explicitly computing finitely atomic representing measures corresponding to moment matrix extensions. Another aspect of this research concerns the interplay between the existence of representing measures and the existence of sum-of-squares representations for positive polynomials; this aspect is directly related to Lasserre's algorithm for polynomial optimization. Another application of this research concerns the development of new minimal cubature rules on classical domains such as the disk or triangle. Broader impacts will include undergraduate training and research projects for science students from underrepresented minorities, and the use of computing, particularly simulations, as an experimental methodology in mathematics and computer science courses.

截断多变量矩问题及其应用：算子理论方法 本研究关注的是多变量截断矩问题的算子理论方法。对于有限的真实的多重序列，我们寻求了在欧氏空间中存在正Borel表示测度的具体充要条件，该测度的幂矩与多重序列的相应元素重合.我们将多序列关联到相应的矩矩阵。众所周知，当且仅当矩矩阵允许扩展到更大的半正定矩矩阵，而这又允许平坦，即，保秩，扩展到更大的矩量矩阵。本文研究了这种矩量矩阵扩张的存在性和最小尺寸。列依赖关系的时刻矩阵确定一个代数品种，其中包含任何代表措施的支持，本研究涉及的代数品种的描述，它是可能建立所需的平坦的时刻矩阵扩展。更一般地说，本研究关注的表示措施，其支持包含在一个规定的半代数闭集。在这种情况下，我们需要平坦扩张（如上所述），其对应于半代数集的局部化矩阵也是半正定的。这一部分的研究涉及半代数（或代数）集，其中正多项式承认度有界加权平方和表示;应用程序直接涉及有限收敛拉瑟尔的多项式优化理论。本研究的另一个方面涉及的算法，明确计算atomic表示措施，这部分研究的应用导致建设的最小或接近最小的多变量立方规则的数值分析。 本研究的目的是在多变量截断矩问题中发展新的双原子表示测度的存在唯一性准则。截断矩问题在诸如算子理论（加权移位的次正态性）、插值理论（经典Nevanlinna-Pick理论）、数值分析（多变量容积规则）、控制理论（信号处理）、优化理论（区域上的多项式优化）和真实的代数几何（正多项式的加权平方和表示）等领域中起着至关重要的作用。本研究的主要重点是一种方法，以多维截断矩问题的基础上的扩展理论的时刻矩阵相关联的时刻数据。当这个矩阵承认一个无限的，积极的，有限的秩矩矩阵的扩展，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对应的扩展的联合频谱上的原子表示措施。本研究的主要目标是确定允许所需扩展的力矩数据的具体条件。本研究还涉及算法显式计算对应的矩矩阵扩展的原子表示措施。本研究的另一个方面涉及之间的相互作用的存在性表示措施和存在的平方和表示正多项式;这方面是直接关系到拉瑟尔的算法多项式优化。这项研究的另一个应用涉及到发展新的最小容积规则的经典领域，如磁盘或三角形。更广泛的影响将包括为来自代表性不足的少数群体的理科学生开展本科生培训和研究项目，以及使用计算，特别是模拟，作为数学和计算机科学课程的实验方法。