Fourier and Fourier-type Algebras of Lie Groups

调和分析是数学的一个基本分支，它解决以下中心问题：如何将函数或信号表示或近似为基波的组合？这种信号表示在当今用于信号存储、传输和降噪的技术中被大量使用。不幸的是，调和分析方法在各种情况下表现不佳，因为它们是基于一系列动作的结果不变的假设而设计的，而不考虑动作的执行顺序。然而，动作通常是不可互换的。例如，在三维(3D)空间中执行连续旋转的结果对旋转的顺序高度敏感。事实上，现实世界是一个非常“不可交换”的宇宙。3D旋转空间是李群的一个简单例子，李群是物理学中自然产生的。本项目将利用泛函分析和Lie理论的技术来推进Lie群的非对易调和分析理论。傅里叶变换及其类似物是经典调和分析的基石。为了将傅里叶变换的概念推广到非阿贝尔群，开创了非对易调和分析的现代领域。这里的广泛哲学是利用群表示和希尔伯特空间上的算子理论来捕捉群的非阿贝尔性质。非对易调和分析的一个主要趋势是研究与非阿贝尔群的傅里叶分析(和表示理论)有关的函数代数。与环境群的正则表示相联系的傅立叶代数是这类函数代数的一个基本例子。这个项目研究了各种类型的局部紧(Lie)群的傅里叶代数及其加权形式的Banach代数行为，特别是导子理论和谱理论。最终目的是研究群的Lie结构与其傅立叶代数的Banach代数性质之间的联系。该项目由数学科学部的分析计划和已建立的激励竞争性研究计划(EPSCoR)共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Harmonic analysis is a fundamental branch of mathematics that addresses the following central problem: How can a function or signal be represented or approximated as a combination of basic waves? Such representations of signals are heavily used in today's technology for storage, transmission, and noise reduction of signals. Unfortunately, harmonic analytic methods perform poorly in various circumstances, as they are designed based on the assumption that the outcome of a series of actions is unchanged, regardless of the order in which the actions are performed. However, actions are often not interchangeable. For example, the result of performing consecutive rotations in three-dimensional (3D) space is highly sensitive to the order of rotations. Indeed, the real world is a very "non-commutative" universe. The space of 3D rotations is a simple example of a Lie group, which arise naturally in physics. This project will use techniques of functional analysis and Lie theory to advance the theory of non-commutative harmonic analysis for Lie groups. Fourier transforms and its analogues are the cornerstone of classical harmonic analysis. To generalize the concept of a Fourier transform to non-Abelian groups, the modern field of non-commutative harmonic analysis was initiated. The broad philosophy here is to employ group representations and the theory of operators on Hilbert spaces to capture the non-Abelian nature of a group. A major trend in non-commutative harmonic analysis is to investigate function algebras related to the Fourier analysis (and representation theory) of non-Abelian groups. The Fourier algebra, which is associated with the regular representation of the ambient group, is a fundamental example of such function algebras. This project investigates Banach algebraic behavior, in particular derivation theory and spectral theory, of the Fourier algebra and its weighted versions for various classes of locally compact (Lie) groups. The ultimate goal is to investigate the connections between the Lie structure of a group and the Banach algebraic properties of its Fourier algebra. This project is jointly funded by the Analysis Program in the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.