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

摘要：本文研究了多变量矩问题的算子理论方法。我们研究的典型问题是多变量截断矩问题：给定一个有限的d维实数序列，寻求d维欧几里德空间上存在一个正Borel测度的具体充要条件，该测度的给定数据表示该测度的连续幂矩。为了研究这种表示测度的存在性，我们将数据与有限矩矩阵联系起来。已知当且仅当矩矩阵允许扩展为无限有限秩正矩矩阵时，存在有限原子表示测度。用最少原子表示的测度对应于最小秩的扩展。我们试图建立这种扩展的具体的充分必要条件，并开发算法来显式计算对应于扩展的表示测度。在矩矩阵为奇异的情况下，我们研究了以下猜想：当且仅当矩矩阵是正的，递归生成的，且矩阵的秩不大于与数据自然相关的代数变化的大小时，存在一个表示测度。这一猜想对于一阶或二阶平面曲线上的矩问题是成立的，因此我们研究了更高阶曲线上的矩问题的猜想。本研究还涉及上述扩展中最小等级的估计；这种估计与某些多项式优化算法的收敛性有关，也与数值分析中最小培养规则的大小有关。本研究的目的是为多变量截断矩问题（其数据对应于连续幂矩直至固定有限度）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（加权位移的次正态性）、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）和优化理论（区域上的多项式优化）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变量截断矩问题的方法。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。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标准。本研究还涉及实现这些标准的算法的发展。一个主要的应用将是为圆盘和三角形等经典域上的测度制定新的最小曲率规则；另一个应用涉及多项式优化算法的收敛性。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。

ABSTRACT This research concerns an operator-theoretic approach to multivariable moment problems. The prototypical problem that we study is the Multivariable Truncated Moment Problem: Given a finite d-dimensional real sequence, we seek concrete necessary and sufficient conditions so that there exists a positive Borel measure on d-dimensional Euclidean space for which the given data represent successive power moments of the measure. To study the existence of such a representing measure, we associate to the data a finite moment matrix. It is known that a finitely atomic representing measure exists if and only if the moment matrix admits an extension to an infinite, finite-rank, positive moment matrix. Representing measures with the fewest atoms correspond to extensions of minimal rank. We seek to establish concrete necessary and sufficient conditions for such extensions, and also to develop algorithms for explicitly computing representing measures corresponding to extensions. In the case when the moment matrix is singular, we study the following conjecture: there exists a representing measure if and only if the moment matrix is positive, recursively generated, and the rank of the matrix is at most equal to the size of the algebraic variety natuarally associated to the data. This conjecture is true for moment problems on planar curves of degree one or two, so we study the conjecture for curves of higher degree. This research also concerns estimates of the minimal rank in the above-mentioned extensions; such estimates are related to the convergence of certain polynomial optimization algorithms and also to the size of minimal 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 (with data corresponding to successive power moments up to a fixed finite degree). Truncated moment problems play essential roles 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), and Optimization Theory (polynomial optimization over a region). The principal focus of this research is an approach to multivariable 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. The primary goal of this research is to determine concrete criteria on the moment data which permit the desired extension. This research also concerns the development of algorithms to implement these criteria. One principal application will be to develop new minimal cubature rules for measures on classical domains such as the disk and triangle; another application concerns the convergence of polynomial optimization algorithms. 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.