Combinatorial Questions in Harmonic Analysis

调和分析中的组合问题

• 批准号: 9801410
• 负责人: Nets Katz
• 金额: $ 7.27万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801410&HistoricalAwards=false
• 关键词: Combinatorial; Questions; Harmonic; Analysis

Proposal: DMS-9801410 Principal Investigator: Nets Katz Abstract: The purpose of this project is to study directional maximal operators in the plane. The major fact underlying this study is the existence of a Besicovitch set, which rules out the boundedness of maximal operators over all directions. When the set of directions is restricted, less is understood. Katz will work on unboundedness in all L^p spaces for maximal operators over Ahlfors regular sets of directions with dimension greater than zero. He will also try to work out Wolff's conjecture about sets that contain a d-dimensional set in each direction. Both of these problems involve finding elasticity in the known constructions of Besicovitch-like sets. A large part of mathematics (as well as physics and signal processing) involves the interaction between time and frequency. In signal processing, time is a one-dimensional continuum and a signal is a function of time. Much depends upon what can be said about the instantaneous frequency of the signal at a given time. Time and frequency also occur as a metaphor in quantum mechanics. Position takes the place of time and momentum the place of frequency. The Heisenberg uncertainty principle limits how instantaneously we may specify a frequency. It says that a note requires a length of time to be heard -- its wavelength. In the quantum mechanical metaphor, there is no need for time -- now called position -- to be one-dimensional. Thus, neither is frequency so constrained. The relevant ranges needed to specify frequencies become rectangles oriented in various directions. To answer questions about this type of universe, it is necessary to understand the interaction between rectangles in different directions. The purpose of this project is to improve that understanding in the setting of two-dimensional space.

提案：DMS-9801410主要研究者：Nets Katz 翻译后摘要：本项目的目的是研究在平面上的方向极大算子。这项研究的主要事实是存在一个贝西科维奇集，它排除了所有方向上的极大算子的有界性。当方向的集合受到限制时，理解的就更少。卡茨将工作的无界性在所有L^p空间的最大运营商超过Ahlfors经常集的方向与尺寸大于零。他还将尝试工作沃尔夫的猜想集包含一个d维集在每个方向。这两个问题都涉及到在已知的Besicovitch类集合的构造中寻找弹性。 数学（以及物理学和信号处理）的很大一部分涉及时间和频率之间的相互作用。在信号处理中，时间是一维连续体，信号是时间的函数。这在很大程度上取决于在给定时间信号的瞬时频率。在量子力学中，时间和频率也是一个隐喻。位置代替时间，动量代替频率。海森伯测不准原理限制了我们可以在多短的时间内确定频率。它说一个音符需要一段时间才能被听到--它的波长。在量子力学的比喻中，时间（现在称为位置）不需要是一维的。因此，频率也不受如此限制。指定频率所需的相关范围变为朝向各个方向的矩形。要回答关于这类宇宙的问题，有必要了解不同方向的矩形之间的相互作用。这个项目的目的是提高在二维空间设置的理解。