Harmonic analysis and Lie groups

摘要：本课题研究现代谐波分析中的各种问题。它结合了与对称空间有关的李群的抽象调和分析和表示理论的思想和问题，以及经典欧几里得调和分析中众所周知的工具和问题。我们的问题列表包括在伪黎曼对称空间的正则表示中出现的一系列表示的详细研究，特别是使用复分析工具的这些表示的几何实现。本研究的一部分是在正则表示中离散发生的特殊函数与表示的球形特性之间的相互作用。我们的研究还包括对称空间的紧化，表示理论在特殊函数中的应用，特别是拉盖尔函数和多项式。另一方面，本文还涉及到小波理论、锥上的函数空间以及其他对称空间，特别是Besov空间的相关问题。提出的工作结合了几个数学领域的方法和思想：复杂分析，流形和函数空间上的群作用，特别是Besov， Fock和Hardy空间，以及经典调和分析。它甚至借鉴了一些应用数学的思想。部分拟议工作将与我们的学生以及美国和欧洲的专家合作完成。谐波分析起源于傅立叶对热方程的研究，这使他考虑将周期函数展开为三角函数的叠加。这既可以解释为常系数微分算子的谱分解，也可以解释为正则表示分解为不可约表示。简而言之，调和分析的主题是通过将函数分解成更简单的函数来研究函数或函数空间。在微分方程理论中，这种分解意味着将任意函数写成特征函数的和或积分。在一些应用中，如在图像处理中，小波显示为用于近似或表示信号的基本原子。如果我们有一个对称群作用于系统，那么我们想把一个任意函数写成在对称群下以简单和可控的方式变换的函数的和，从而导致对称群的表示理论。这两个方面通常涉及对积分变换和相应核函数的研究。

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.