Aspects of thinness in harmonic analysis

调和分析中的稀疏性方面

• 批准号: 44597-2011
• 负责人: Hare, Kathryn
• 金额: $ 1.31万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568928
• 关键词: Aspects; thinness; harmonic; analysis

Harmonic analysts seek to develop mathematical theories that help to find solutions to problems that might have arisen from mathematical physics, electrical engineering and other branches of mathematics. My research in harmonic analysis is motivated by the harmonic analysis version of the Uncertainty Principle. This principle is philosophically related to the Heisenberg Uncertainty Principle in that, roughly speaking, it says information gained about a function has to be 'paid for' by a corresponding loss of control on its Fourier transform, or vice versa. This is significant in applications. For instance, it implies that one cannot have a radio signal that is bounded in both time and frequencies. In my research, I develop rigorous and quantitative interpretations of this principle. I study functions/ operators/measures that are "small", in some sense, and seek to understand the consequences of this. 1. Lie groups are often used in physics as they describe real-world geometry. Orbital measures are elementary components in this setting and have small support. The first part of my project is to understand the relationship between the size of their transforms and the associated geometric and algebraic structures. This is important for analysis on Lie groups. 2. Sidon sets can be defined in terms of the analytic properties of the functions whose Fourier transform is supported on the set. Although they are small in an intuitive sense, they have a complicated structure. In the second part of the project our objective is to characterize Sidon sets in terms of sets that are simpler to understand. 3. The third part of the project is to use tools from fractal analysis to study the geometry of natural measures that are associated with Cantor sets. These are of interest because of their pathological behaviour. Surprising geometry has been discovered in simple examples. Our goal is to determine whether this odd behaviour is wide spread and why it is occurring.

谐波分析者寻求发展数学理论，帮助找到数学物理、电子工程和其他数学分支中可能出现的问题的解决方案。