Almost Periodic and Multivariable Periodic Matrix Functions: Extensions, Factorizations, Applications

准周期和多变量周期矩阵函数：扩展、因式分解、应用

• 批准号: 9800704
• 负责人: Leiba Rodman
• 金额: $ 12.05万
• 依托单位: College of William and Mary
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2000-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800704&HistoricalAwards=false
• 关键词: Almost; Periodic; Multivariable; Matrix; Functions

DMS-9800704 PI: L. Rodman, I. M. Spitkovsky, H. J. Woerdeman. For periodic matrix functions of several variables, the positive extension and contractive extension problems will be studied.The positive extension problem arises in signal processing when spectral estimation of stochastic processes are required. The contractive counterpart arises in such fields as the worst-case control, e.g., in the model-matching problem. New support sets with the positive extension property will be identified, and a description of all solutions (including a maximum entropy solution) of the respective extension problems will be given. For this purpose, new versions of the band method, suited for the multivariable setting and including dual norm results, will be formulated. For finite sets which do not have positive extension property, numerical algorithms based on the reduction to matrix completion problems will be developed. The results obtained will be related back to their applications.The principal investigators will continue a systematic study of almost periodic factorization. This factorization arises naturally in diffraction theory, distributed parameters control systems, and the spectral estimation problem for a class of stochastic processes. New existence results will be obtained and then used in applications to almost periodic versions of the Corona problem and stable rank one property, Krein's theorem on orthogonal matrix polynomials, the Beurling-Lax theorem and the geometry of shift-invariant subspaces of the Besicovitch space. Almost periodic factorization will also be used in the spectral theory of various operators (including Toeplitz, Wiener-Hopf and other convolution type operators) with matrix almost periodic and semi almost periodic symbols on weighted Lebesgue spaces, and in almost periodic extensions and interpolation problems. Connections with the respective problems for periodic functions of several variables will be exploited. New procedures for explicit almost periodic factorization will be developed and applied to a constructive solution of convolution type (in particular, difference) equations on finite intervals. These results will be further applied in the scattering theory for the generalized Schrodinger equation. The proposed research aims to develop new mathematical tools and techniques, motivated by potential uses in diverse areas of science and technology. A typical application in medical sciences, seismology, signal processing, or physics involves reconstruction (as much as available data allows) of the intrinsic structure of an object of study, given only a few surface measurements. Statistical applications involve prediction, or prognosis, of a process in probabilistic terms, when only a brief history is known. There are currently available mathematical techniques that underlie many recent technological advances in these areas. However, the existing techniques deal mostly with the case when there is only one variable (typically time or distance along a line) involved. The immense complexity of real-life systems calls for new mathematical developments, in particular it calls for an extension of the existing theory to the case of more variables (allowing for instance for planar and spatial variables). The proposed research seeks to make substantial contributions in this direction. Interaction and collaboration with many researchers in different fields is an integral part of the proposal. The principal investigators also plan to involve undergraduate students in parts of their research.

DMS-9800704 PI：L。罗德曼岛M. Spitkovsky，H. J·沃德曼 对于多元周期矩阵函数， 本文将研究信号处理中随机过程谱估计的正延拓问题。收缩对应物出现在 诸如最坏情况控制的字段，例如，在模型匹配问题上。 将识别具有正扩展属性的新支持集， 以及所有解的描述（包括最大熵解） 将给出相应的扩展问题。为此，新 带方法的版本，适用于多变量设置， 包括双范数结果。对于有限集合， 不具有正延拓性质，将发展基于矩阵完备化问题的数值算法。所得到的结果将与它们的应用联系起来。主要研究者将继续对概周期分解进行系统的研究。这种分解在衍射理论、分布参数控制系统和一类随机过程的谱估计问题中自然出现。将获得新的存在性结果，然后将其应用于几乎周期形式的Corona问题和稳定的秩一性质，正交矩阵多项式的Krein定理，Beurling-Lax定理和 Besicovitch空间的平移不变子空间的几何。 几乎周期因子分解也将用于谱 各种算子理论（包括Toeplitz，Wiener-Hopf和其他 卷积型算子），在加权Lebesgue空间上和概周期空间中， 扩展和插值问题。的联系 多元周期函数的相关问题 将被利用。新的程序显式几乎周期因式分解将开发和应用于有限区间上的卷积型（特别是差分）方程的建设性解决方案。这些结果将进一步应用于广义薛定谔方程的散射理论。 拟议的研究旨在开发新的数学工具， 技术，在不同领域的潜在用途的动机， 科学技术一个典型的应用， 医学、地震学、信号处理或物理学涉及 重建（尽可能多的可用数据允许）的内在 研究对象的结构，只给出几个表面测量。 统计学的应用包括对一个过程的预测或预测 从概率的角度来看，当我们只知道一个简短的历史时。 目前有可用的数学技术， 这些领域的许多最新技术进步。但 现有的技术主要处理只有一个 变量（通常是沿着沿着的时间或距离）。的 现实生活系统的巨大复杂性要求新的数学 发展，特别是它要求延长现有的 理论的情况下，更多的变量（允许例如平面和 空间变量）。这项研究旨在使实质性的 在这个方向上的贡献。与许多人的互动和协作 不同领域的研究人员是该提案的组成部分。的 主要研究人员还计划让本科生参与 他们的部分研究。