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Harmonic Analysis Homogeneous Spaces and its Applications

调和分析齐次空间及其应用

基本信息

批准号：
15540182
负责人：
KUMAHARA Keisaku
金额：
$ 1.66万
依托单位：
The University of the Air
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

We studied the properties of the Fourier transforms on homogeneous spaces of Lie groups. Especially we focused our attention on the sampling theorem of Shannon and its generalizations. The harmonic analysis on homogeneous spaces has developed as extension of the classical Fourier analysis on a real number line or a plane. The Plancherel theorem and the Paley-Wiener theorem have built the important flow in it. These theorems characterize the images by the Fourier transform of some function spaces. In the case of the real line, the Paley-Wiener theorem can be considered from its self-duality to be the theorem by which functions with Fourier transform image of compact support, that is, band-limited functions, are characterized. Such a function is restriction to the real axis of an exponential type entire function. It is the Shannon sampling theorem that the values in all points of such function are with the values at the discrete points on a real axis. This theorem has played the decisive role in communication theory. In this research, we mainly dealt with the problem of the econstruction of the function by the sampling of the Radon conversion on symmetrical spaces.We obtained the following results :(1)A sampling formula on the sphere. (2)A sampling formula for Radon transform on the real hyperbolic space, (3)A reconstruction formula on real hyperbolic space using discrete samples of Radon transform, (4)A sampling formula on the complex sphere, (5)A sampling formula for Radon transform on complex hyperbolic space, (6)A reconstruction formula on complex hyperbolic spaces using discrete samples of Radon transform, (7)A reconstruction formula on Riemannian symmetric spaces by using discrete samples of Radon transform, (8)A sampling formula for the Fourier-Jacobi series, (9)A sampling formula on complex semisimple Lie groups.

研究了李群齐性空间上的傅里叶变换的性质。特别是对Shannon采样定理及其推广进行了研究。齐次空间上的调和分析是经典Fourier分析在真实的数线或平面上的推广。Plancherel定理和Paley-Wiener定理建立了它的重要流，这些定理通过对某些函数空间的傅立叶变换来刻画象。在真实的直线的情形下，Paley-Wiener定理可以从其自对偶性被认为是刻画具有紧支集的傅里叶变换像的函数，即带限函数的定理。这样的函数是对指数型整函数的真实的轴的限制。Shannon采样定理是这样的函数的所有点的值与真实的轴上的离散点的值相同。这个定理在通信理论中起了决定性的作用。本文主要研究了对称空间上Radon变换的抽样构造函数的问题，得到了如下结果：（1）球面上的抽样公式。(2)A真实的双曲空间上的Radon变换的采样公式，（3）利用Radon变换的离散样本在真实的双曲空间上重建公式，（4）复球面上的采样公式，（5）复双曲空间上的Radon变换的采样公式，（6）利用Radon变换的离散样本在复双曲空间上重建公式，（7）用Radon变换的离散样本重构黎曼对称空间的一个公式，（8）Fourier-Jacobi级数的一个采样公式，（9）复半单李群上的一个采样公式。