Harmonic Analysis Homogeneous Spaces and its Applications

研究了李群齐次空间上傅里叶变换的性质。我们特别关注了香农的抽样定理及其推广。齐次空间上的调和分析是对经典傅里叶分析在实数线或平面上的推广。Plancherel定理和Paley-Wiener定理建立了其中的重要流程。这些定理通过一些函数空间的傅里叶变换来表征图像。在实线的情况下，Paley-Wiener定理可以从其自对偶性考虑为具有紧支持的傅里叶变换像的函数，即带限函数的表征定理。这样的函数是对指数型整函数实轴的约束。该函数所有点的值与实轴上离散点的值一致，这是香农抽样定理。这个定理在传播理论中起了决定性的作用。在本研究中，我们主要通过对称空间上Radon转换的采样来处理函数的构造问题。得到了以下结果：(1)球面上的采样公式。(2)实双曲空间上Radon变换的采样公式，(3)实双曲空间上Radon变换的离散样本重构公式，(4)复球面上的采样公式，(5)复双曲空间上Radon变换的采样公式，(6)复双曲空间上Radon变换的离散样本重构公式，(7)riemann对称空间上Radon变换的离散样本重构公式，(8) Fourier-Jacobi级数的抽样公式；(9)复半单李群的抽样公式。

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.