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

摘要 本研究关注的是多变量矩问题的算子理论方法。我们研究的原型问题是多变量截断矩问题：给定一个有限的d维真实的序列，我们寻求具体的必要和充分条件，使存在一个积极的Borel措施d维欧氏空间的给定的数据表示连续的权力的时刻的措施。为了研究这样一个代表措施的存在，我们将数据关联到一个有限矩矩阵。已知一个双原子表示测度存在当且仅当矩矩阵允许一个无限的、有限秩的、正矩矩阵的扩张。用最少的原子表示测度对应于最小秩的扩张。我们寻求建立具体的必要和充分条件，这样的扩展，并开发算法，明确计算代表措施相应的扩展。在矩矩阵奇异的情况下，我们研究了如下猜想：存在表示测度当且仅当矩矩阵是正的，递归生成的，并且矩阵的秩至多等于与数据自然相关的代数簇的大小.这个猜想对于一次或二次平面曲线上的矩量问题是成立的，因此我们研究了高次曲线上的猜想。本研究还涉及上述扩展中的最小秩的估计;这种估计与某些多项式优化算法的收敛性有关，也与数值分析中的最小体积规则的大小有关。 本研究的目的是开发新的存在性和唯一性准则的多变量截断矩问题（与数据对应的连续的权力时刻到一个固定的有限度）的原子表示措施。截断矩问题在诸如算子理论（加权移位的次正规性）、插值理论（经典Nevanlinna-Pick理论）、数值分析（多变量体积规则）、控制理论（信号处理）和优化理论（区域上的多项式优化）等领域中起着重要作用。本研究的主要重点是一种方法，多变量截断矩问题的基础上的扩展理论的矩矩阵相关联的时刻数据。当这个矩阵允许一个无限的，正的，有限秩矩矩阵的扩展，这种方法产生一个显式的公式的一个原子表示措施。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标准。本研究还涉及算法的发展，以实现这些标准。一个主要的应用将是开发新的最小容积规则的措施，经典领域，如磁盘和三角形;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。

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.