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.

沃尔德曼的提议在于多变量复分析和多变量算子理论的结合。研究的背景是算子值的 Schur 类及其子类 Schur?Agler 类。虽然后者由于 Agler 的开创性工作而得到了更多的理解，但多变量 Schur 类在很大程度上仍未被探索。所提出的程序旨在获得对多变量 Schur 类函数的结构和二变量 Christoffel?Darboux 公式的高维类似物的结构的新颖见解。这项研究的动机是其最终目标，即描述具有酉膨胀的收缩交换元组类，获得多变量 Schur 类中 Nevanlinna?Pick 插值问题的可解性标准，并开发此类函数的系统实现理论。要使用的工具包括散射系统机制（在 PI 上建立动力？最近的工作）和多变量 Toeplitz 算子的 Schur 补集技术（并行开发多变量 Toeplitz 矩阵的快速反演/求解器数值算法）。该程序的主题之一是最佳近似几何问题，与保尔森关于多变量算子值线性冯诺依曼不等式中最佳常数的猜想的一个重要特例相关。该项目解决了多变量插值和因式分解问题的活跃领域中的几个问题。这些问题与当前科学和工程的各个领域相关，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、压缩传感和量子计算。该项目的主要教育组成部分是研究生的监督和本科生的指导，本科生将得到本科生研究经验计划的支持。拟议研究的结果将在多个层面上传播：通过出版物和在国内和国际专业会议上的演讲，其中一些会议也将由其他领域的研究人员参加，例如计算机科学、物理学和工程学；通过正式和非正式的教育活动，包括每周在德雷克塞尔大学举办的分析研讨会，由 PI 主办并由教职员工和学生参加；通过与其他机构的同事进行访问交流进行合作；通过 PI？网站和预印本服务器。