Harmonic Analysis and Affinely Invariant Measures

谐波分析和仿射不变测量

• 批准号: 9986804
• 负责人: Daniel Oberlin
• 金额: $ 4.46万
• 依托单位: Florida State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-05-15 至 2003-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9986804&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Affinely; Invariant; Measures

AbstractThis project will focus on the relationship of affinely-invariant measures tocertain problems in harmonic analysis. Drury observed that affine arclengthappears to be the natural measure to use when considering averaging and Fourierrestriction operators associated with certain curves. More recently, the PI hasobtained certain uniform results along these lines for curves in two dimensionsand for surfaces in higher dimensions. He intends to continue theseinvestigations and also to consider some questions of an integral-geometricnature. The latter stem from the concept of affine dimension, an analog ofHausdorff dimension which also takes into account a set's curvature.The averaging operators considered in this project belong to a class of smoothingoperators which play an important role in the study of physical processes likewave motion and heat propagation. These operators are also related to the Radontransforms which appear in tomography. Similar operators are important in signalanalysis and communications theory. The Fourier restriction operators consideredhere are part of the theory of Fourier analysis. Their study contributes to thegeneral understanding of the Fourier transform, a tool which is vital toapplications of mathematics ranging from electrical engineering andcommunications theory through fluid dynamics.

摘要本课题主要研究谐波分析中若干问题与仿射不变测度的关系。德鲁里观察到，当考虑与某些曲线相关的平均和傅里叶限制算子时，仿射弧长似乎是使用的自然度量。最近，对于二维曲线和高维曲面，PI沿着这些线得到了某些一致的结果。他打算继续这些研究，并考虑一些积分几何性质的问题。后者源于仿射维数的概念，仿射维数是豪斯多夫维数的类比，它也考虑了集合的曲率。本课题所考虑的平均算子属于一类光滑算子，在波动和热传播等物理过程的研究中起着重要作用。这些算子也与层析成像中出现的radontransform有关。相似算子在信号分析和通信理论中很重要。这里考虑的傅里叶限制算子是傅里叶分析理论的一部分。他们的研究有助于对傅里叶变换的普遍理解，傅里叶变换是一种对从电气工程和通信理论到流体动力学的数学应用至关重要的工具。