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

多变量截断矩问题及其应用：算子理论方法研究了多变量截断矩问题的算子理论方法。对于有限实多序列，我们寻求欧几里得空间中正Borel表示测度存在的具体充分必要条件，该测度的幂矩与多序列的相应元素一致。我们给多序列关联一个相应的矩矩阵。我们知道，当且仅当矩矩阵允许扩展到更大的正半定矩矩阵时，一个表示测度存在，而这个正半定矩矩阵又允许一个平坦的，即保持秩的，扩展到更大的矩矩阵。本文研究了此类矩矩阵扩展的存在性和最小尺寸问题。矩矩阵中的列依赖关系决定了一个包含任何表示测度支持的代数变体，本文研究了可能建立期望的平面矩矩阵扩展的代数变体的描述。更一般地说，本研究关注的是支持度包含在规定的半代数闭集中的表示测度。在这种情况下，我们需要平面扩展（如上所述），其中对应于半代数集的定位矩阵也是正半定的。这部分的研究涉及半代数（或代数）集，其中正多项式允许有度有界的加权平方和表示；Lasserre多项式优化理论的应用直接关系到有限收敛性。该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了数值分析中最小或近最小多变量培养规则的构建。本研究的目的是为多变量截断矩问题的有限原子表示测度建立新的存在唯一性准则。截断矩问题在算子理论（加权移位的次正态性）、插值理论（经典的Nevanlinna-Pick理论）、数值分析（多变量立方体规则）、控制理论（信号处理）、优化理论（区域上的多项式优化）和实代数几何（正多项式的加权平方和表示）等领域中发挥着重要作用。本研究的主要焦点是基于矩矩阵与矩数据相关联的扩展理论的多维截断矩问题的解决方法。当这个矩阵允许无限的、正的、有限秩矩矩阵扩展时，这种方法得到了一个有限原子表示测度的显式公式，该测度支持在与该扩展相对应的正常元组算子的联合谱上。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。本研究还涉及显式计算与矩矩阵扩展相对应的有限原子表示测度的算法。本研究的另一个方面涉及表征测度的存在与正多项式的平方和表征的存在之间的相互作用；这方面与Lasserre的多项式优化算法直接相关。本研究的另一个应用涉及在诸如圆盘或三角形等经典域上发展新的最小培养规则。更广泛的影响将包括为来自代表性不足的少数民族的理科生提供本科培训和研究项目，以及在数学和计算机科学课程中使用计算，特别是模拟，作为一种实验方法。

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.