Topics in Fourier Analysis

傅立叶分析主题

• 批准号: 9970042
• 负责人: Andreas Seeger
• 金额: $ 15.9万
• 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: 1999
• 资助国家: 美国
• 起止时间: 1999-05-15 至 2003-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970042&HistoricalAwards=false
• 关键词: Topics; Fourier; Analysis

Proposal: DMS-9970042Principal Investigator: Andreas SeegerAbstract: The proposed research will be concerned with a number of topics in Fourier analysis related to singular integral operators, Fourier integral operators with degenerate canonical relations, Radon transforms, averages over curves and surfaces and associated maximal functions, regularity of wave propagation, spectral multipliers on noncompact manifolds, and the multitude of interrelations between these topics.The subject of Fourier analysis has provided powerful tools for various areas of mathematics and its applications. The P.I. proposes to develop further and refine some of these tools. This should lead to a better qualitative understanding of integral operators that arise in the study of partial differential equations from mathematical physics, as well as in the study of integral geometry and tomography. The potential concrete implications of improving this understanding are enormous. Tomography, for instance, not only provides the mathematical foundation for the reconstruction methods that are employed constantly in medical imaging but also furnishes the key to interpreting seismic data, lending it a "real world" significance that cannot be overstated.

提案：DMS-9970042原理研究者：Andreas Segerabstrabstract：拟议的研究将与与奇异积分运算符，傅里叶傅立叶型运算符有关的傅立叶分析中的许多主题，具有退化的规范关系，ra rave，ra不变，ra不变，曲线和表面上的平均和相关的跨度，跨度的跨度，远距离的跨度，近视效应，远距离跨度，远距离繁殖，远距离繁殖，远距离效应，远距离繁殖范围，远距离跨度，远距离跨度，远距离效能，远距离繁殖率，远距离效能，远距离繁殖范围，这些主题之间的相互关系。傅立叶分析的主题为数学及其应用的各个领域提供了强大的工具。 P.I.建议进一步发展并完善其中一些工具。这应该导致对整体运算符的更好的定性理解，这是在研究数学物理学的部分微分方程以及整体几何和整体批摄影研究中出现的。改善这种理解的潜在具体含义是巨大的。例如，断层扫描不仅为不断用于医学成像中的重建方法提供了数学基础，而且还为解释地震数据提供了关键，使其成为“现实世界”的重要性，无法被夸大。