Aspects of Thinness in Harmonic Analysis

谐波分析中的稀度方面

• 批准号: RGPIN-2016-03719
• 负责人: Hare, Kathryn
• 金额: $ 1.6万
• 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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=614575
• 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, in part, by the harmonic analysis version of the Uncertainty Principle, a guiding philosophy loosely related to the Heisenberg Uncertainty Principle. 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 qualitative and quantitative interpretations of this principle. I study functions, measures and sets that are thin or small, in some sense, and my goal is to understand the consequences of this. This topic has long been of interest to mathematicians for it is well known that ‘thin’ objects are important in many problems and can exhibit interesting phenomena. For example, Weierstrass’ famous example of a continuous, nowhere differentiable function was a trigonometric series with thin support of its transform, and the Cantor set, a thin but uncountable set, is important in many branches of mathematics. Recently, there have been important connections found relating thinness ideas from harmonic analysis with other areas of mathematics such as number theory and combinatorics.My research program will have two major components. 1. Thin sets: Sidon sets can be defined in terms of analytic properties of the functions whose Fourier transform is supported on the set. Although they are thin in an intuitive sense, their structure is complicated. The objective of this part of our program is to characterize Sidon sets in terms of thinner sets that are simpler to understand, a fundamental and long-standing problem in harmonic analysis. 2. Thinly supported measures: (a) The classical Cantor measure is a well-known example of a measure which has both small support and (somewhat) small Fourier transform. It is of interest throughout mathematics because of its pathological behavior and yet tractability. In the second part of the program we will develop techniques to quantify the local behavior of Cantor-like measures and other measures that arise from an iterative construction, but have overlap. These are important, current topics in fractal geometry.(b) Lie groups are often used in physics as they describe real-world geometry. Orbital measures are elementary components in this setting and like Cantor measures have small support and small transform. The final part of my research program is to understand the smoothness properties of this class of thin measures and their associated geometric structures. I anticipate this will have important applications for harmonic analysis on Lie groups.

谐波分析师寻求发展数学理论，帮助找到可能出现的问题的解决方案，从数学物理，电气工程和其他数学分支。我在调和分析方面的研究部分是由不确定性原理的调和分析版本激发的，这是一种与海森堡不确定性原理松散相关的指导哲学。粗略地说，它说，关于函数的信息必须通过傅立叶变换的相应控制损失来“支付”，反之亦然。这在应用中具有重要意义。例如，它意味着一个人不能拥有一个在时间和频率上都有限制的无线电信号。在我的研究中，我对这一原则进行了定性和定量的解释。我研究函数、度量和集合，在某种意义上，是薄的或小的，我的目标是理解这一点的后果。这个话题一直感兴趣的数学家，因为它是众所周知的，“薄”的对象是重要的，在许多问题，可以表现出有趣的现象。例如，维尔斯特拉斯著名的例子，一个连续的，无处可微的功能是一个三角级数薄支持其变换，和康托集，薄，但不可数集，是重要的许多分支的数学。最近，已经发现了重要的联系，从调和分析与数学的其他领域，如数论和组合学相关的瘦的想法。1.薄集：西顿集可以根据其傅里叶变换在集合上得到支持的函数的解析性质来定义。虽然它们在直觉上很薄，但它们的结构很复杂。我们计划的这一部分的目的是表征西顿集的更薄的集合，更容易理解，调和分析中的一个基本和长期存在的问题。2.支持薄弱的措施：（a）经典康托测度是一个著名的例子，它既有小支撑又有（有点）小的傅立叶变换。它是整个数学的兴趣，因为它的病态行为，但易处理。在该计划的第二部分中，我们将开发技术来量化Cantor样的措施和其他措施，从迭代建设产生的局部行为，但有重叠。这些都是分形几何中重要的当前主题。(b)李群经常用于物理学中，因为它们描述了真实世界的几何。轨道测度是这类情形的基本组成部分，并且像康托测度一样具有小的支撑和小的变换。我的研究计划的最后一部分是了解这类薄措施和他们的相关几何结构的光滑性。我预计这将有重要的应用调和分析李群。