Topics in Fourier Analysis

傅立叶分析主题

• 批准号: 1200261
• 负责人: Andreas Seeger
• 金额: $ 33.3万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1200261&HistoricalAwards=false
• 关键词: Topics; Fourier; Analysis

This mathematics research project is concerned with several problems in harmonic analysis. A primary focus is on the problem of estimating operators that commute with translation and rotations. It is desirable to prove effective characterizations for the boundedness of such operators in Lebesgue spaces. Such results and the techniques to prove them can be used to investigate regularity properties of the solutions of classical equations in mathematical physics, such as the wave equation, the Schroedinger equation and other dispersive equations. Seeger and his collaborators will also study various questions about the behavior of the Fourier transform on submanifolds of Euclidean space. Such problems are tied to quantitative questions about eigenfunctions of differential operators on compact manifolds. Other projects concern the regularity properties of averaging operators, the boundedness of singular integral operators, generalized Radon transforms, and various pointwise convergence questions related to differentiation problems and Fourier series.Methods and techniques from Fourier analysis have always found wide applications in understanding physical phenomena in the natural sciences and in engineering. The proposed investigations of the solutions of wave and Schroedinger equations, as well as some of the projects on singular integral operators, are motivated by questions from physics. The proposed research on generalized Radon transforms is relevant to problems that arise in medical imaging. This research project also has an educational component, as Seeger will direct Ph.D. students and mentor postdoctoral researchers both in their research and in teaching. Seeger will also continue to be involved in the organization of conferences to facilitate interactions of researchers and students in the mathematics field of analysis.

本课题研究谐波分析中的几个问题。一个主要的焦点是估计与平移和旋转交换的算子的问题。这是理想的证明有效的特征，在Lebesgue空间中的有界性的此类算子。这些结果及其证明方法可用于研究数学物理中经典方程解的正则性，如波动方程、Schroedinger方程和其它色散方程。西格和他的合作者还将研究各种问题的行为傅里叶变换的子流形的欧几里得空间。这样的问题是绑紧微分算子的特征函数的紧流形上的定量问题。其他项目涉及的正则性平均运营商，奇异积分运营商的有界性，广义Radon变换，以及各种点态收敛问题有关的微分问题和傅立叶series.Methods和技术傅立叶分析一直发现广泛的应用在理解物理现象在自然科学和工程。波动方程和薛定谔方程的解的拟议调查，以及奇异积分算子的一些项目，是由物理问题的动机。广义Radon变换的研究与医学成像中出现的问题有关。这个研究项目也有一个教育的组成部分，因为西格将直接博士学位。学生和导师博士后研究人员在他们的研究和教学。西格还将继续参与会议的组织，以促进数学分析领域研究人员和学生的互动。