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: 1304208
• 负责人: Mishko Mitkovski
• 金额: $ 5.04万
• 依托单位: Clemson University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-16 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1304208&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.

这个项目中的大部分研究活动本质上可以看作是压缩算子半群的摄动问题。对这些问题的拟议处理可以简要描述如下。一个表示Hilbert空间，压缩半群在其上充当一个子空间链，使得在每个子空间上，这个半群几乎充当一个酉群。相应的谱图产生了一系列解析函数空间。然后，我们的目标是通过研究这些空间以及它们与由某个模型半群生成的空间之间的关系来理解半群的性质。后者往往归结为某些Toeplitz算子的可逆性问题。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算子的内射性问题的强大方法。像调和分析中常见的情况一样，希尔伯特变换的微妙性质应该再次发挥核心作用。本项目提出的大多数问题都属于调和分析领域。调和分析的核心思想是将复杂的信号表示为更简单的信号的组合--原子在某种意义上对手头的问题是规范的。由于测量的不精确性，人们有时需要使用略有不同的原子系统，它可能具有也可能不具有正则系统的理想性质。需要进行仔细的分析，以确定理想测量下的属性在多大程度上继续保持在现实世界的情况下。该项目的很大一部分用于进一步开发这种分析所需的数学工具。在信号处理和控制理论领域可能有潜在的应用。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数分析的经典领域，从而使未来几代数学家以及其他科学家更容易接触到这些技术。作为一所主要的科技大学的一员，调查员还将纳入其中的一些新想法，为新一代STEM专业的本科生和研究生提供最新的、高质量的教育。