Decompositions for multivariable Schur-class functions, Christoffel-Darboux type formulas, and related problems

多变量 Schur 类函数、Christoffel-Darboux 类型公式的分解以及相关问题

• 批准号: 0901628
• 负责人: Hugo Woerdeman
• 金额: $ 47.56万
• 依托单位: Drexel University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901628&HistoricalAwards=false
• 关键词: Decompositions; multivariable; Schur; class; functions

WoerdemanThe proposal lies at the interface of multivariable complex analysis and multivariable operator theory. The setting of study is the operator-valued Schur class and its subclass, the Schur?Agler class. While the latter is more understood due to Agler?s seminal work, the multivariable Schur class remains largely unexplored. The proposed program is aimed to gain a novel insight into the structure of multivariable Schur-class functions and higher-dimensional analogs of the two-variable Christoffel?Darboux formula. The investigation is motivated by its ultimate goals which would be to describe the class of commuting tuples of contractions having unitary dilations, to obtain solvability criteria for the Nevanlinna?Pick interpolation problem in the multivariable Schur class, and to develop a theory of system realizations for this class of functions. Tools to be employed include the machinery of scattering systems (building momentum on PIs? recent work) and the technique of Schur complements of multivariable Toeplitz operators (with a parallel development of fast inversion/solver numerical algorithms for multivariable Toeplitz matrices). One of the themes in this program is a best approximation geometric problem, tied to an important special case of Paulsen?s conjecture on the best constant in the multivariable operator-valued linear von Neumann inequality.The project addresses several questions in the active area of multivariable interpolation and factorization problems. These questions are of current relevance to a variety of areas in science and engineering, which include, but are not limited to, system and control theory, filter design, signal and image processing, compressive sensing, and quantum computation. The main educational component of the project is the supervision of graduate students and the mentoring of undergraduates who will be supported by the Research Experiences for Undergraduates program.The results of the proposed research will be disseminated at severallevels: through publications and presentations at national and international professional meetings, some of which will also be attended by researchers from other fields, such as computer science, physics and engineering; via formal and informal educational activities, including the weekly Analysis seminar at Drexel University run by the PIs and attended by both faculty and students; via visit exchanges with colleagues from other institutions for collaboration purposes; via the PIs? web sites and preprint servers.

沃德曼（Woerdeman）提案在于多变量复杂分析和多变量运算符理论的界面。研究的设置是运营商价值的Schur类及其子类Schur？​​Agler类。尽管由于Agler的开创性工作，后者更加理解，但多变量的Schur类基本上仍未得到探索。提出的计划旨在获得对多变量Schur级函数的结构和两种可变的基督教“ darboux公式”的较高维度类似物的新见解。该调查是由其最终目标的动机，即描述具有单一膨胀的收缩的通勤类别，以获取Nevanlinna的可溶性标准？在多变量SCHUR类中选择插值问题，并为此功能开发系统实现的系统理论。要使用的工具包括散射系统的机械（在PIS上建立动量？最近的工作）以及多变量Toeplitz操作员的Schur补充技术（以及快速反转/求解器数值算法的平行开发，用于多变量toeplitz矩阵）。该程序中的一个主题之一是最佳近似几何问题，与Paulsen的重要特殊情况有关，这是在多变量操作员价值的线性线性von Neumann不平等中的最佳常数。该项目解决了多个可互动插入和分支化问题的活性领域中的几个问题。这些问题与科学和工程学的各个领域具有当前的相关性，其中包括但不限于系统和控制理论，滤波器设计，信号和图像处理，压缩感应和量子计算。该项目的主要教育组成部分是研究生的监督以及对本科生的研究经验的指导，他们将在本科生计划的研究经验中支持。拟议的研究的结果将在几项级别上进行：通过国内和国际专业会议的出版物和演讲，其中一些领域的研究人员也会由其他领域的研究人员送达，例如计算机科学和工程学，物理学和工程学，物理学和工程学;通过正式和非正式的教育活动，包括由PIS经营的Drexel University的每周分析研讨会，并由教职员工和学生参加；通过访问与其他机构的同事进行协作的交流；通过pis？网站和预印式服务器。