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

PI: L.罗德曼，I. M.斯皮特科夫斯基，H. J.沃尔德曼。对于多变量周期矩阵函数，研究其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，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，例如在模型匹配问题中。将识别具有正可拓性质的新支持集，并给出相应可拓问题的所有解（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对偶范数结果的新版本波段法。对于不具有正可拓性质的有限集，将发展基于约简到矩阵补全问题的数值算法。所获得的结果将与它们的应用联系起来。主要研究人员将继续对几乎周期性分解进行系统研究。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。将得到新的存在性结果，并将其应用于Corona问题的概周期版本和稳定秩一性质、关于正交矩阵多项式的Krein定理、Beurling-Lax定理以及Besicovitch空间的移不变子空间的几何。几乎周期分解还将用于加权Lebesgue空间上具有矩阵几乎周期和半几乎周期符号的各种算子（包括Toeplitz， Wiener-Hopf和其他卷积型算子）的谱理论，以及几乎周期扩展和插值问题。将利用与多变量周期函数的各自问题的联系。新的显式几乎周期分解程序将被开发并应用于有限区间上的卷积型（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，其动机是在不同的科学和技术领域的潜在用途。在医学、地震学、信号处理或物理学中的典型应用涉及（在现有数据允许的范围内）重构研究对象的内在结构，仅给出少量的表面测量值。统计应用包括在只知道一个短暂历史的情况下，以概率的方式对一个过程进行预测或预测。目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，现有的技术主要处理只有一个变量（通常是时间或沿直线的距离）的情况。现实系统的巨大复杂性需要新的数学发展，特别是需要将现有理论扩展到更多变量的情况下（例如允许平面和空间变量）。拟议的研究力求在这方面作出重大贡献。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。

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.