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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=658960
• 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.**

该提案包括有关现代复杂和功能分析，矩阵分析及其在其他数学和工程领域的应用中的基本问题的几个相互关联的任务，例如近似理论，数学物理学，控制理论，信号处理和电气工程。分析功能空间和作用于它们的操作员一直是研究的活跃领域。 RKHS提供了一种现代而有力的工具，可以查看经典和新的问题，因此它们在应用和纯科学的众多领域中起着重要作用。复制核的出现可以追溯到几位著名的数学家的创始作品，例如Nevanlinna，Pick和Schur，以精确的约束插值。从那时起，RKHS在研究广泛空间的性能中的核心作用和强度证明了这一点，这是通过突破性的结果证明的，取决于插值，采样，独特性和不变的子空间，由Aleman，Carleson，Carleson，Fricain，Ransford，Ransford，Ransford，Richter，Richter，Richter，Sarason，Seip等。内核。******我们考虑了几个这样的空间，例如Hardy，Dirichlet，Bergman，Model和De Branges-Rovnyak空间。这些空间上最著名的运营商是前进和向后移动操作员。这些对象会导致更一般的概念，例如Toeplitz，Hankel Operators，Berezin Transform and Composition Operators。任何此类操作员都可以解释为作用在环境空间中功能系数形成的序列空间上的无限尺寸矩阵。为了治疗无限的尺寸矩阵，我们自然会考虑它们的截断，因此经典矩阵理论显示了它的面孔。因此，从这个角度看，矩阵理论的技术（无限尺寸和有限维度）在RKHS中应用。繁殖内核家族的几何特性，例如完整性，最小性，是riesz的基础或渐进性基础，与全体形态功能空间中插值，采样和独特性等属性密切相关。我们在这里主要对Hardy，Dirichlet和模型空间及其概括De Branges-Rovnyak空间感兴趣。这导致我们研究了Dirichlet和de Branges-Rovnyak空间以及Hardy空间模型子空间中的独特集和零集，循环性以及插值和采样序列。它们在光谱理论，广义强化空间，矩阵倒置的规范控制和控制理论中具有自然应用。此外，我们遇到的问题本身就在矩阵理论的主题中就很有趣。一个著名的问题，即一个旧猜想的延续，是双重型矩阵特征值的基因座。**