【作者】 刘沛；

【导师】 何建勋；

【作者基本信息】 广州大学 ， 应用数学， 2007， 硕士

【摘要】 令H₁是3-维的海森堡群, H1上径向函数空间的基础流形记为[0,+∞)×R,称为Laguerre超群（[25]）.自然地, Kⁿ = [0,+∞)ⁿ×Rⁿ称为乘积Laguerre超群.本文首先建立了Kⁿ = [0,+∞)ⁿ×Rⁿ上的平移算子和L2（Kⁿ,dμ）上的Plancherel公式.其次,讨论Kⁿ = [0,+∞)ⁿ×Rⁿ上的广义小波变换和Radon变换理论.然后,我们构造S（Kⁿ）（施瓦茨空间）的一个特征子空间SR（Kⁿ）,指出Radon变换在SR（Kⁿ）上是一一映射,并给出与SR（Kⁿ）等价的S（Kⁿ）的另一个特征子空间S?,2（Kⁿ）.最后,在弱意义下,利用广义小波逆变换得到Kⁿ = [0,+∞)ⁿ×Rⁿ上Radon变换的逆公式.类似地,此结果在海森堡群上成立.更多还原

【Abstract】 Let H1 be the 3-dimensional Heisenberg group. The fundamental manifold ofthe radial function space for H1 can be denoted by [0, +∞)×R, which is just theLaguerre hypergroup(see[25]). Naturally, K~n = [0, +∞)~n×R~n is product Laguerrehypergroup. In this paper, we construct a generalized translation operator on K~n =[0,+∞)~n×R~n, and establish the Plancherel formula on L2(K~n,dμ). Then we discussthe continuous wavelets transform and Radon transform on K~n, and we characterizea subspace SR(K~n) of S (K~n)(Schwartz space), on which Radon transform is abijection. Also, we give another characterized subspace S?,2(K~n) which is equivalentto SR(K~n). Using the inverse wavelet transform, we obtain an inversion formulaof Radon transform on K~n in the weak sense. Analogously, the same case can beextended to the Heisenberg group.更多还原