Workshop on Harmonic Analysis and Applications

谐波分析及应用研讨会

• 批准号: 0637383
• 负责人: Gestur Olafsson
• 金额: $ 1.76万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-11-15 至 2007-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0637383&HistoricalAwards=false
• 关键词: Workshop; Harmonic; Analysis; Applications

AbstractOlafssonThis project deals with a variety of problems in modern harmonic analysis.It combines ideas and problems from abstract harmonic analysisand representation theory on Lie groups related tosymmetric spaces with tools and questions wellknown from classical Euclidean harmonic analysis. Our list ofproblems includes a detailed study of series of representations occurringin the regular representation on Pseudo-Riemannian symmetric spaces and, inparticular, geometric realizations of those representations usingtools from complex analysis. Part of this study is the interplay betweenspecial functions and the spherical character of representationsoccurring discretely in the regular representation.Our study also includes compactificationof symmetric spaces, application of representation theory to special functions,in particular, Laguerre functions and polynomials.On the other hand the proposal includes problems related to wavelet theory,function spaces on cones and other symmetric spaces, in particular,Besov spaces. The proposed work combines methods and ideas from several areasof mathematics: Complex analysis, group actions on manifolds and functionspaces, in particular, Besov, Fock, and Hardy spaces, and classical harmonic analysis. It even borrows some ideas from applied mathematics. Parts of the proposed work will bedone in collaboration with our students as well as specialists in USA and Europe.Harmonic analysis has its origin in the work of Fourier on the heat equation,which led him to consider the expansion of a periodic functions into superpositionof trigonometric functions. This can be interpreted either as the spectraldecomposition of the differential operators with constant coefficients, oras decomposition of regular representation into irreducible representations.In short, the subject of harmonic analysis is to study functions orfunction spaces by decomposing the functions into simpler functions. Inthe theory of differential equations this decomposition means to writean arbitrary functions as a sum or integral of eigenfunctions. In severalapplications, as in image processing, the wavelets shows up as the basicatoms used to approximate or represent the signal. If we have a symmetrygroup acting on the system, then we would like to write an arbitraryfunction as a sum of functions that transforms in a simple and controllableway under the symmetry group, leading to representation theory ofthe symmetry group. Both aspects usually involve the study of integraltransforms and the corresponding kernel function.

Abstractolafssonthis这项项目在现代谐波分析中处理了各种问题。它结合了摘要和谐波分析的思想和问题，与谎言组相关的tosymmore -tosymmortric空间以及工具和工具和问题结合了古典欧几里得和谐分析中众所周知的问题。我们的问题列表包括一项详细的研究，详细研究出现在伪riemannian对称空间上的定期表示形式，以及在复杂分析中使用工具对这些表示的内部几何实现。这项研究的一部分是在定期代表性中离散地分散表现的表达功能的相互作用和领域特征。我们的研究还包括对称空间的紧凑，将代表理论应用于特殊功能，尤其是laguerre函数和多项官方的特殊功能。另一方面，该提案与symere symeres相关的范围以及其他范围的范围以及其他范围，以及其他范围。提出的工作结合了数学几个领域的方法和思想：复杂的分析，尤其是对多种功能空间的小组行动，尤其是besov，fock，hardy空间，以及经典的谐波分析。它甚至从应用数学借用了一些想法。拟议的工作的某些部分将与我们的学生以及美国和欧洲的专家合作。HarmonicAnalysis起源于傅立叶在热方程式上的工作，这使他考虑了定期功能扩展到三角函数的叠加。这可以被解释为具有恒定系数的差分运算符的光谱分解，而是将定期表示形式分解为不可减少表示形式。简而言之，谐波分析的主题是通过将函数分解为简单函数来研究功能机能空间。在微分方程理论中，这种分解意味着将任意函数作为本征函数的总和或整体。在几种贴法中，如图像处理中，小波显示为用于近似或表示信号的基本变性。如果我们在系统上具有对称组，那么我们想编写一个任意功能作为函数的总和，该功能在对称组下在简单可控制的道路中转换，从而导致对称群的表示理论。这两个方面通常都涉及整合性的研究和相应的内核函数。