Harmonic Analysis and Affinely Invariant Measures

摘要本课题将重点研究仿射不变测度与调和分析中某些问题的关系。德鲁里观察到，仿射弧度似乎是在考虑与某些曲线相关的平均和傅立叶限制算子时使用的自然度量。最近，对于二维曲线和高维曲面，PI沿着这些直线得到了某些一致的结果。他打算继续这些研究，并考虑一些具有积分几何性质的问题。后者源于仿射维度的概念，它是Hausdorff维度的一种模拟，它也考虑了集合的曲率。本项目中考虑的平均算子属于一类光滑算子，在研究波动和热传播等物理过程中发挥着重要作用。这些算符也与层析成像中出现的Radon变换有关。类似的运算符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总体理解，傅里叶变换是一种对从电气工程、通信理论到流体力学等数学应用至关重要的工具。

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.