Hilbert spaces of analytic functions and their applications

解析函数的希尔伯特空间及其应用

• 批准号: 1101251
• 负责人: Mishko Mitkovski
• 金额: $ 10.19万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2012-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101251&HistoricalAwards=false
• 关键词: Hilbert; spaces; analytic; functions; their

Most of the proposed research activities in this project can be viewed essentially as perturbation problems for contractive operator semigroups. The proposed treatment of these questions can be briefly described as follows. One represents the Hilbert space on which the contractive semigroup acts as a chain of subspaces so that on each of the subspaces the semigroup acts almost as a group of unitaries. The corresponding spectral picture produces a chain of spaces of analytic functions. The goal is then to understand the properties of the semigroup by investigating these spaces and their relationship to the spaces generated by a certain model semigroup. The latter often reduces to a problem about invertibility properties of certain Toeplitz operators. In its essence, the method that the investigator suggests to treat these problems can be viewed as a delicate form of the classical argument principle. It has in its basis the powerful method of N. Makarov and A. Poltoratski regarding the injectivity problem for Toeplitz operators. As is often the case in harmonic analysis, the delicate properties of the Hilbert transform again should play the central role.The majority of the problems proposed in this project lie within the area of harmonic analysis. The core idea of harmonic analysis is the possibility of representing complicated signals as a combination of simpler signals - atoms which are in a sense canonical for the problem at hand. Due to the imprecision of measurements one is sometimes required to use a slightly different system of atoms, which may or may not possess the ideal properties of the canonical system. Careful analysis is required to determine to which extent the properties present under ideal measurements continue to hold in a real world situation. A large part of this project is devoted to the further development of the mathematical tools necessary for such an analysis. Potential applications are possible in the areas of signal processing and control theory. In addition, another goal of this project is to popularize the classical areas of harmonic and complex analysis by softening some existing deep techniques, thus making them more accessible to the future generations of mathematicians as well as to other scientists. As a member of a major science-technology university, the investigator will also incorporate some of these new ideas to offer an up-to-date, quality education of the new generations of STEM majors, at both the undergraduate and graduate level.

该项目中提出的大多数研究活动本质上可以被视为收缩算子半群的扰动问题。这些问题的拟议处理可简要描述如下。一个代表希尔伯特空间，在该空间上，收缩半群充当子空间链，因此在每个子空间上，半群几乎充当一组酉群。相应的光谱图产生了解析函数的空间链。然后，目标是通过研究这些空间及其与某个模型半群生成的空间的关系来理解半群的属性。后者通常会简化为有关某些托普利茨算子的可逆性的问题。从本质上讲，研究者提出的处理这些问题的方法可以被视为经典论证原则的一种微妙形式。它的基础是 N. Makarov 和 A. Poltoratski 关于 Toeplitz 算子的单射性问题的强大方法。正如调和分析中经常出现的情况一样，希尔伯特变换的微妙特性应该再次发挥核心作用。该项目中提出的大多数问题都属于调和分析领域。谐波分析的核心思想是将复杂信号表示为更简单信号（原子）的组合的可能性，这些信号在某种意义上是当前问题的规范。由于测量的不精确性，有时需要使用稍微不同的原子系统，该系统可能具有也可能不具有规范系统的理想特性。需要仔细分析以确定理想测量下呈现的属性在多大程度上在现实世界的情况下继续保持不变。该项目的很大一部分致力于进一步开发此类分析所需的数学工具。在信号处理和控制理论领域有潜在的应用。此外，该项目的另一个目标是通过软化一些现有的深层技术来普及调和和复分析的经典领域，从而使未来几代数学家以及其他科学家更容易理解它们。作为一所主要科技大学的成员，研究者还将融入其中一些新理念，为新一代 STEM 专业的本科生和研究生提供最新的优质教育。