Functional transforms and their kernels on symmetric spaces and Lie groups

对称空间和李群上的函数变换及其核

• 批准号: 170653-2006
• 负责人: Sawyer, Patrice
• 金额: $ 0.87万
• 依托单位: Laurentian University of Sudbury
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=361173
• 关键词: Functional; transforms; their; kernels; symmetric

A symmetric space refers, in plain words, to the kind of "universe" I am studying. The general context of my work is harmonic analysis of this universe. The basic idea of harmonic analysis, itself a generalization of Fourier analysis, is to represent general functions in terms of more basic ones, i.e. spherical functions. Historically, such studies concerned spaces such as the real line (the one-dimensional space); the plane (the two-dimensional universe); or space (the three-dimensional universe). A natural extension was then the Euclidean n-dimensional space. There are more complicated objects mathematicians want to work with (the sphere being a natural example). Symmetric spaces are sufficiently complex objects to be of interest as a generalization, while sufficiently well-behaved to make their study relevant and interesting (not to mention accessible!). Not only can one consider questions such as the heat diffusion on a symmetric space, but it also makes sense to study probability theory using such objects. As well, the behaviour of the basic functions i.e. the spherical functions becomes important. Natural assumptions as to how a well-known phenomenon in the Euclidean setting will behave in a different situation are either confirmed or contradicted. These questions can be addressed by finding explicit formulae for the spherical functions (when possible) or by using general principles.

对称空间，通俗地说就是指我正在研究的那种“宇宙”。我工作的总体背景是对这个宇宙的调和分析。调和分析的基本思想本身是傅里叶分析的推广，是用更基本的函数（即球函数）来表示一般函数。从历史上看，此类研究涉及空间，例如实线（一维空间）；平面（二维宇宙）；或空间（三维宇宙）。一个自然的扩展就是欧几里得 n 维空间。数学家想要处理更复杂的物体（球体就是一个自然的例子）。对称空间是足够复杂的对象，作为概括而令人感兴趣，同时其表现也足够良好，使他们的研究具有相关性和趣味性（更不用说易于理解了！）。人们不仅可以考虑对称空间上的热扩散等问题，而且利用此类物体来研究概率论也很有意义。同样，基本函数（即球函数）的行为也变得很重要。关于欧几里得环境中的众所周知的现象在不同情况下如何表现的自然假设要么被证实，要么被矛盾。这些问题可以通过找到球函数的明确公式（如果可能）或使用一般原理来解决。