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. Rodman、I. M. Spitkovsky、H. J. Woerdeman。 对于多变量周期矩阵函数，将研究正可拓和收缩可拓问题。在信号处理中，当需要对随机过程进行谱估计时，就会出现正可拓问题。收缩对应物出现在最坏情况控制等领域，例如模型匹配问题。 将识别具有正可拓性质的新支持集，并给出各个可拓问题的所有解（包括最大熵解）的描述。为此，将制定适用于多变量设置并包括双范数结果的新版本带法。对于不具有正扩展性质的有限集，将开发基于矩阵完成问题简化的数值算法。获得的结果将与其应用相关。主要研究人员将继续对几乎周期因式分解进行系统研究。这种因式分解自然出现在衍射理论、分布式参数控制系统和一类随机过程的谱估计问题中。将获得新的存在结果，然后将其应用于 Corona 问题的几乎周期版本和稳定的一级属性、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。 准周期因式分解还将用于加权勒贝格空间上具有矩阵准周期和半准周期符号的各种算子（包括 Toeplitz、Wiener-Hopf 和其他卷积类型算子）的谱理论，以及准周期扩展和插值问题。将利用与多个变量的周期函数的相应问题的联系。将开发显式几乎周期因式分解的新程序，并将其应用于有限间隔上卷积型（特别是差分）方程的构造解。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和技术，其动机是在不同科学和技术领域的潜在用途。医学、地震学、信号处理或物理学中的典型应用涉及在仅给出少量表面测量的情况下重建研究对象的内在结构（在可用数据允许的范围内）。 统计应用涉及在仅了解简短历史的情况下以概率术语对过程进行预测或预测。 当前可用的数学技术是这些领域许多最新技术进步的基础。然而，现有技术主要处理仅涉及一个变量（通常是时间或沿线的距离）的情况。现实生活系统的巨大复杂性需要新的数学发展，特别是它需要将现有理论扩展到更多变量的情况（例如允许平面和空间变量）。拟议的研究旨在朝这个方向做出重大贡献。与不同领域的许多研究人员的互动和合作是该提案的一个组成部分。主要研究人员还计划让本科生参与他们的部分研究。