Mathematical Sciences: Problems in Linear Analysis

提案编号：DMS-9500924提案标题：线性分析中的问题PI/PD：Leiba Rodman CO-PI/PD：Hugo J.Woerdeman摘要主要研究人员计划研究数学科学中高度活跃的研究领域中的各种相互关联的问题，该领域涉及算符理论、矩阵分析及其在科学和技术中的应用，还使用与理论和应用数学密切相关的领域：函数论、傅立叶分析。除了在其他主题上继续工作外，本研究的主要焦点如下：1.矩阵和算子值函数的内插问题；2.有限和无限矩阵和算子的完备化；3.Riccati方程；4.Krein空间算子；5.应用。主要研究人员努力开发新的结果，这些结果可以在抽象的数学背景下证明(例如，在*-代数方面)，同时在各种更具体的环境中产生新的和有用的信息。主要调查人员计划将新获得的抽象结果与最初的应用领域联系起来。提案中强调的应用包括先进制造(工程控制系统)和高性能计算和通信(断层成像)，这两个领域已被确定为对国家需求有影响的两个战略领域。其他相关的应用领域包括：工程控制系统、偏振光光学、量子化学中的动能最小化、波的衍射、信号处理和模式识别。计划与不同领域的许多研究人员(数学、计算机科学、工程学)、美国以外的不同国家(加拿大、荷兰、意大利、以色列、挪威、法国、英国)以及工业和政府科学家(国家卫生研究所、NEC研究所)进行互动，并构成该项目的组成部分。

PROPOSAL NO.: DMS - 9500924 PROPOSAL TITLE: Problems in Linear Analysis PI/PD: Leiba Rodman CO-PI/PD: Hugo J. Woerdeman ABSTRACT The principal investigators plan to study a variety of interconnected problems in a highly active area of research in Mathematical Sciences that involves operator theory, matrix analysis, and their applications in science and technology, using also closely related areas of pure and applied mathematics: function theory, Fourier analysis. In addition to continuing work on other topics, specific problems are outlined in the following subjects which are the main focus of the research: 1. Interpolation problems for matrix and operator valued functions; 2. Completions of finite and infinite matrices and operators; 3. Riccati equations; 4. Krein space operators; 5.Applications. The principal investigators strive to develop new results that can be proved in an abstract mathematical context (e. g., in terms of *-algebras), and that at the same time yield new and useful information in various more specific settings. The principal investigators plan to relate the newly obtained abstract results to the original areas of application. Applications that are emphasized in the proposal include Advanced Manufacturing (engineering control systems) and High Performance Computing and Communications (tomography), which have been identified as two of the strategic areas with impact on national need. Other areas of relevant application are: engineering control systems, optics of polarized light, kinetic energy minimization in quantum chemistry, diffraction of waves, signal processing, and pattern recognition. Interactions with many researchers in different fields (mathematics, computer science, engineering), various countries besides the U. S. (Canada, The Netherlands, Italy, Israel, Norway, France, United Kingdom), and scientists in industry and government (the National Institute of Health, NEC Research Institute) are planned and form an integral part of this project.