Aspects of Thinness in Harmonic Analysis

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。我在谐波分析方面的研究部分是由不确定性原理的谐波分析版本驱动的，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，它的意思是，一个函数获得的信息必须通过对其傅里叶变换失去相应的控制来“补偿”，反之亦然。这在应用程序中很重要。例如，它意味着一个人不可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。我研究的函数，度量和集合在某种意义上是薄的或小的，我的目标是理解它的结果。这个话题长期以来一直引起数学家的兴趣，因为众所周知，“薄”对象在许多问题中都很重要，并且可以展示有趣的现象。例如，Weierstrass著名的连续的，无处可微的函数的例子是一个三角级数，它的变换支持很弱，而Cantor集合，一个很薄但不可数的集合，在数学的许多分支中都很重要。最近，从调和分析中发现了与其他数学领域如数论和组合有关的重要联系。我的研究计划有两个主要部分。1. 瘦集：西顿集可以用在集合上支持傅里叶变换的函数的解析性质来定义。虽然它们在直观上很薄，但它们的结构却很复杂。我们计划的这一部分的目的是用更容易理解的更薄的集来描述西顿集，这是谐波分析中一个基本和长期存在的问题。2. 稀疏支持测度：(a)经典的康托测度是一个众所周知的例子，它具有小的支持和（有点）小的傅里叶变换。整个数学领域都对它感兴趣，因为它的病态行为和可操控性。在计划的第二部分，我们将开发技术来量化cantor度量和其他度量的局部行为，这些度量来自迭代构造，但是有重叠。这些都是分形几何中重要的、当前的主题。(b)物理学中经常使用李群，因为它们描述了现实世界的几何形状。轨道测度是基本分量，和康托测度一样，支持度和变换都很小。我研究计划的最后一部分是了解这类薄测度及其相关几何结构的平滑特性。我预计这将对李群的谐波分析有重要的应用。

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.