Reproducing Kernel Hilbert Spaces, Matrix Theory, their relations and applications

再现核希尔伯特空间、矩阵理论、它们的关系和应用

• 批准号: RGPIN-2018-04534
• 负责人: Mashreghi, Javad
• 金额: $ 2.04万
• 依托单位: Université Laval
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=682739
• 关键词: Reproducing; Kernel; Hilbert; Spaces; Matrix

This proposal consists of several interrelated tasks on fundamental problems in modern complex and functional analysis, matrix analysis and their applications to other fields of mathematics and engineering, e.g., approximation theory, mathematical physics, control theory, signal processing and electrical engineering. Analytic Function Spaces and the operators acting on them has been an active domain of research. RKHS provide a modern and powerful tool to look at such problems, classic and new, and thus they play an important role in numerous domains of applied and pure sciences. The advent of reproducing kernels goes back to the founding works of several prominent mathematicians like Nevanlinna, Pick, and Schur on exact constrained interpolation. Since then, RKHS made evidence of their central role and strength in the study of properties of a wide range of spaces as demonstrated by breakthrough results on interpolation, sampling, uniqueness, and invariant subspaces by Aleman, Carleson, Fricain, Ransford, Richter, Sarason, Seip, etc. The solution in 2013 of the Feichtinger conjecture is a milestone and opens new research directions in the field of reproducing kernels.******We consider several such spaces, e.g., Hardy, Dirichlet, Bergman, Model and de Branges-Rovnyak spaces. The most celebrated operators on these spaces are the forward and backward shift operators. These objects lead to more general concepts like Toeplitz, Hankel operators, Berezin transform and composition operators. Any such operator can be interpreted as an infinite dimensional matrix acting on the sequence space formed with the coefficients of functions in the ambient space. To treat infinite dimensional matrices, we naturally consider their truncations and thus the classical matrix theory shows its face. Hence, looking from this angle, techniques of matrix theory (infinite dimensional as well as finite dimensional) are applied in RKHS. Geometric properties of families of reproducing kernels like completeness, minimality, being a Riesz basis or an asymptotically orthonormal basis, are intimately related to properties like interpolation, sampling, and uniqueness in spaces of holomorphic functions. We are mainly interested here in Hardy, Dirichlet and model spaces and their generalization de Branges-Rovnyak spaces. This leads us to study uniqueness sets and zero sets, cyclicity, and interpolating and sampling sequences in Dirichlet and de Branges-Rovnyak spaces as well as in model subspaces of Hardy spaces. They have natural applications in spectral theory, generalized Hardy spaces, norm control of matrix inversion, and control theory. Moreover, we encounter questions which are interesting in their own right in the subject of matrix theory. A celebrated question, which is the continuation of an old conjecture, is the loci of eigenvalues of doubly-stochastic matrices.**

该提案包括几个相互关联的任务，涉及现代复变函数分析、矩阵分析及其在数学和工程其他领域的应用等基本问题。近似理论、数学物理、控制理论、信号处理和电气工程。解析函数空间及其上的算子一直是一个活跃的研究领域。RKHS提供了一个现代和强大的工具来看待这些问题，经典的和新的，因此他们在应用和纯科学的许多领域发挥着重要作用。再生核的出现可以追溯到Nevanlinna，Pick和Schur等几位着名数学家在精确约束插值方面的基础工作。从那时起，RKHS证明了他们在研究各种空间的性质方面的核心作用和力量，正如Aleman，Carleson，Ransford，Ransford，Richter，Sarason，Seip，2013年Feichtinger猜想的解决是一个里程碑，并在再生核领域开辟了新的研究方向。我们考虑几个这样的空间，例如，哈代，Dirichlet，Bergman，Model和de Branges-Rovnyak空间。这些空间中最著名的操作符是向前和向后移位操作符。这些对象导致更一般的概念，如Toeplitz，汉克尔算子，Berezin变换和复合算子。任何这样的算子都可以被解释为作用于由周围空间中的函数的系数形成的序列空间的无限维矩阵。在处理无穷维矩阵时，我们自然要考虑它们的截断，从而使经典矩阵理论显露出它的面目。因此，从这个角度来看，矩阵理论（无限维以及有限维）的技术应用在RKHS。再生核族的几何性质，如完备性、极小性、作为Riesz基或渐近正交基，与全纯函数空间中的插值、采样和唯一性等性质密切相关。我们主要感兴趣的是哈代，Dirichlet和模型空间及其推广的de Branges-Rovnyak空间。这导致我们研究的唯一性集和零集，循环性，插值和采样序列在Dirichlet和de Branges-Rovnyak空间以及在模型子空间的哈代空间。它们在谱理论、广义哈代空间、矩阵求逆的范数控制和控制理论中有着自然的应用。此外，我们遇到的问题是有趣的，在自己的权利，在主题的矩阵理论。一个著名的问题是双随机矩阵特征值的轨迹，它是一个古老猜想的延续。