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

截断多元矩问题及其应用：一种算子理论方法这项研究是关于多变量截断矩问题的一种算子理论方法。对于有限实多重序列，我们寻找了欧氏空间中存在正Borel表示测度的具体充要条件，即使该测度的功率矩与多重序列的相应元素重合的测度。我们将相应的矩矩阵与多序列相关联。众所周知，当且仅当矩矩阵允许对更大的半正定矩矩阵进行扩展时，表示度量存在，而半正定矩矩阵又允许对更大的矩矩阵进行平坦的、即保序的扩展。这项研究涉及这种矩矩阵扩张的存在性和最小规模。矩矩阵中的列依赖关系决定了一个代数簇，该代数簇包含任何表示度量的支持度，并且本研究涉及对其可以建立期望的平坦矩矩阵扩张的代数簇的描述。更广泛地说，这项研究涉及支持度包含在规定的半代数闭集内的表示测度。在这种情况下，我们需要(如上所述)与半代数集对应的局部化矩阵也是正半正定的平坦扩张。这部分研究涉及半代数(或代数)集，其中正多项式允许次数有界的加权平方和表示；其应用直接涉及Lasserre多项式优化理论中的有限收敛。这项研究的另一个方面涉及显式计算有限原子表示度量的算法；这部分研究的应用导致在数值分析中构造最小或接近最小的多变量立方体规则。本研究的目的是建立多元截断矩问题中有限原子表示测度的新的存在唯一性准则。截断矩问题在算子理论(加权移位的次正规性)、插值理论(经典的Nevanlinna-Pick理论)、数值分析(多变量尺度规则)、控制理论(信号处理)、最优化理论(区域上的多项式优化)和实代数几何(正多项式表示为加权平方和)等领域中发挥着重要的作用。这项研究的主要焦点是基于与矩数据相关的矩矩阵的扩展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。这项研究的主要目标是确定允许所需扩展的时刻数据的具体条件。这项研究还涉及显式计算对应于矩矩阵扩张的有限原子表示度量的算法。这项研究的另一个方面涉及表示测度的存在性和正多项式的平方和表示的存在性之间的相互作用；这一方面与拉瑟尔的多项式优化算法直接相关。这项研究的另一个应用涉及在圆盘或三角形等经典区域上发展新的最小立方规则。更广泛的影响将包括为来自代表性不足的少数族裔的理科学生进行本科培训和研究项目，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。

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.