Some Problems in Analysis

分析中的一些问题

• 批准号: 1160680
• 负责人: Daniel Oberlin
• 金额: $ 14.06万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-07-15 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1160680&HistoricalAwards=false
• 关键词: Some; Problems; Analysis

This mathematics research project will support the investigation of several important problems in mathematical analysis. One of these problems is in Fourier analysis and concerns the restrictions of Fourier transforms to curves in Euclidean spaces. The other problems in this project are in geometric measure theory. These fall into two categories. The problems in the first of these two categories are all specific instances of the following: what can be said about the dimension of a set in d-dimensional space if that set is known to be the union of a particular class of sets of a certain type. The most famous of such problems is the so-called Kakeya conjecture that a set containing a unit line segment in each possible direction must have full dimension. These problems lead directly to the study of the boundedness of operators known as Radon transforms. The problems in the second of the two above-mentioned categories have to do with the extent to which certain given patterns can occur in sets of a certain fixed size. A prototypical example here is the unit distance problem. That problem asks about the number of line segments of length one which can be formed by joining pairs of points chosen from a given set.This mathematics research project in the areas of Fourier analysis and geometric measure theory has the potential to impact a quite diverse collection of science and technology-related disciplines. Here are some examples: Fourier analysis is an important tool in the study of wavelets, and these in turn have applications ranging from data compression and image analysis to communications theory; the study of the restriction theory of Fourier transforms has applications to disciplines that are concerned with waves and wave-like phenomena, such as fluid mechanics and quantum physics; the Radon transforms are the basic mathematical tools which make possible today's sophisticated medical imaging techniques; sets called fractals, which are one of the objects of study in geometric measure theory, provide patterns for the antennae in some cellular telephones.

这个数学研究项目将支持数学分析中几个重要问题的研究。这些问题之一是在傅立叶分析和关注的限制傅立叶变换曲线在欧几里德空间。这个项目中的其他问题是在几何测量理论。这些问题分为两类。这两类中的第一类问题都是以下问题的具体实例：如果已知d维空间中的一个集合是某类特定类型集合的并集，那么该集合的维数是什么？其中最著名的问题是所谓的挂谷猜想，即在每个可能的方向上包含单位线段的集合必须具有全维。这些问题直接导致研究的有界性的运营商被称为拉东变换。上述两类问题中的第二类问题与某些特定模式在某种固定规模的集合中出现的程度有关。这里的一个典型例子是单位距离问题。这个问题是关于从给定集合中选择的点对连接起来可以形成的长度为1的线段的数量。这个数学研究项目在傅立叶分析和几何测量理论领域有可能影响各种各样的科学和技术相关学科。以下是一些示例：傅立叶分析是小波研究中的重要工具，这些小波又具有从数据压缩和图像分析到通信理论的应用范围;傅立叶变换的限制理论的研究具有与波和类似波的现象有关的学科的应用，例如流体力学和量子物理学; Radon变换是使当今复杂的医学成像技术成为可能的基本数学工具;称为分形的集合是几何测量理论中的研究对象之一，其为某些蜂窝电话中的天线提供图案。