Topics in Fourier Analysis

傅立叶分析主题

• 批准号: 0200186
• 负责人: Andreas Seeger
• 金额: $ 37.5万
• 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: 2002
• 资助国家: 美国
• 起止时间: 2002-06-15 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200186&HistoricalAwards=false
• 关键词: Topics; Fourier; Analysis

Proposal Number: DMS-0200186PI: Andreas SeegerABSTRACTThe proposed research will be concerned with severalinterrelated topics in Fourier analysis. In particularit is proposed to work on precise regularity propertiesof oscillatory integral and Fourier integral operatorswith degenerate canonical relations. The mappingproperties are governed by the geometry of the canonicalrelations. Here we are, in particular, interested inthe cases that come up when studying averaging operatorsassociated to curves in higher dimensions. Other projects include the behavior of singular maximalfunctions and rough singular integrals on classes ofintegrable functions, the mapping properties of waveoperators on nilpotent groups, and questions concerningthe failure of weak amenability of Lie groups.The research project is focused on some fundamentalquestion in Fourier analysis, with an emphasis on estimates for various oscillatory integral operators.Such estimates are in particular crucial forunderstanding the qualitative and quantitative behaviorof Radon transforms and related Fourier integraloperators. These operators arise in various areas ofmathematics, such as partial differential equations,complex analysis and integral geometry, as well as innon-mathematical applications such as tomography andmedical imaging. As an example we mention that X-raytomography involves inverting a Radon transformoperator; i.e. one seeks to determine an image fromknowledge about the averages of a function for aprescribed family of lines. The operators involved ininverting this restricted X-ray transform tend to beFourier integral operators as considered in thisproposal.

提案编号：DMS-0200186PI：Andreas Seegerabstractthe拟议的研究将与傅立叶分析中的多个与众不同的主题有关。提议尤其提议在振荡积分和傅立叶积分运算符的精确规律性属性上工作，并具有归化规范关系的傅立叶积分。地图杂质受规范关联的几何形状的控制。在这里，特别是，我们对研究平均操作员的情况下的情况感兴趣，以较高的维度曲线。 其他项目包括奇异最大函数的行为和对综合功能类别的奇异积分，绘制spoperator在nilpotent群体上的绘制属性，以及有关谎言群体弱化性的失败的问题。该研究项目集中于傅立叶分析中的一些基本问题，并强调各种仪式的估计，以估算各种整体运营商，以估算整体的运算群体。 ra的定性和定量行为转换和相关的傅立叶整体分类器。这些操作员出现在各个数学领域，例如部分微分方程，复杂的分析和积分几何形状，以及诸如层析成像和媒体成像等内数学应用。例如，我们提到的是X射线术涉及反转ra transforperer。即，一个人试图从知识中确定图像，以了解Aprescipt的线条家庭的函数平均值。涉及对这种受限X射线变换的操作员往往会像本权中所考虑的那样使整体操作员变得更加不可或缺。