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: $ 4.08万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759467
• 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.

本建议包括几个相互关联的任务，涉及现代复杂和泛函分析、矩阵分析及其在其他数学和工程领域（如近似理论、数学物理、控制理论、信号处理和电气工程）中的基本问题。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提供了一个现代而强大的工具来研究这些问题，无论是经典的还是新的，因此它们在应用科学和纯科学的许多领域发挥着重要作用。复制核的出现可以追溯到Nevanlinna、Pick和Schur等几位杰出数学家关于精确约束插值的奠基工作。从那时起，RKHS在广泛空间性质研究中的核心作用和实力得到了证明，如Aleman、Carleson、Fricain、Ransford、Richter、Sarason、Seip等人在插值、采样、唯一性和不变子空间方面的突破性成果。2013年费希廷格猜想的解决是一个里程碑，为核的再现领域开辟了新的研究方向。我们考虑了几个这样的空间，如Hardy， Dirichlet, Bergman， Model和de Branges-Rovnyak空间。这些空间上最著名的运算符是正向移位和向后移位运算符。这些对象引出了更一般的概念，如Toeplitz、Hankel算子、Berezin变换和复合算子。任何这样的算子都可以解释为作用于由环境空间中函数的系数构成的序列空间上的无限维矩阵。在处理无穷维矩阵时，我们自然会考虑它们的截断，从而经典矩阵理论露出了它的面目。因此，从这个角度来看，矩阵理论技术（无限维和有限维）在RKHS中的应用。复制核族的几何性质，如完备性、极小性、Riesz基或渐近标准正交基，与全纯函数空间中的插值、抽样和唯一性等性质密切相关。我们主要对哈代、狄利克雷和模型空间以及它们在布朗日-洛夫尼亚克空间中的推广感兴趣。这导致我们研究Dirichlet和de Branges-Rovnyak空间以及Hardy空间的模型子空间中的唯一性集和零集、循环性、插值和采样序列。它们在谱理论、广义Hardy空间、矩阵逆范数控制和控制理论中都有很好的应用。此外，在矩阵理论的主题中，我们遇到的问题本身就很有趣。一个著名的问题是双随机矩阵的特征值的轨迹，它是一个老猜想的延续。