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Harmonic analysis on Grassmann manifolds and its applications to Radon transforms and inverse problems

格拉斯曼流形的调和分析及其在 Radon 变换和反演问题中的应用

基本信息

批准号：
16540136
负责人：
KAKEHI Tomoyuki
金额：
$ 2.37万
依托单位：
University of Tsukuba
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

In this research project, we studied the following (1), (2) and (3).(1) Dual Radon transforms on affine Grassmann manifolds.(2) Moment conditions and support theorems for Radon transforms on affine Grassmann manifolds.(3) Range characterization of the matrix Radon transform.(1) The main result is as follows. Let G(d,n) be the affine Grassmann manifold of d-dimensional planes in the n-dimensional Euclidian space. We assume that q<p and dim(G(p,n))<dim(G(q,n)). Let R be the Radon transform from the space of smooth functions on G(p,n) to that on G(q,n). Then the range of the Radon transform R is characterized by the system of Pfaffian equations.(2) The main result is as follows. We assume that p<q and dim(G(p,n))=dim(G(q,n)). The Radon transform R associated with the inclusion incidence relation maps the Schwartz space on G(p,n) to that on G(q,n). Let f be a Schwartz class function on G(p,n). If the image Rf is compactly supported, then the function f is also compactly supported. In addition, we proved that the range of R is characterized by generalized moment conditions.(3) The main result is as follows. Let M be the space of n×k matrices, and let Ξ be the space of matrix planes in M. The matrix Radon transform from functions on M to functions on Ξ is defined as the integral of a function on each matrix plane. Then the range of the matrix radon transform is characterized as the kernel of a generalized Pfaffian type operator arising from the corresponding Cartan motion group.

在本研究项目中，我们研究了以下（1）、（2）和（3）。(1)仿射格拉斯曼流形上的对偶Radon变换。(2)仿射Grassmann流形上Radon变换的矩条件和支撑定理。(3)Radon变换矩阵的值域特征。(1)主要结果如下。设G（d，n）是n维欧氏空间中d维平面的仿射格拉斯曼流形。我们假设q<p且dim（G（p，n））<dim（G（q，n））。设R是G（p，n）上光滑函数空间到G（q，n）上光滑函数空间的Radon变换。然后，Radon变换R的范围由Pfidian方程组表征。(2)主要结果如下。我们假设p<q且dim（G（p，n））=dim（G（q，n））。与包含关联关系相关联的Radon变换R将G（p，n）上的Schwartz空间映射到G（q，n）上的Schwartz空间。设f是G（p，n）上的Schwartz类函数.如果图像Rf是紧支持的，那么函数f也是紧支持的。此外，我们还证明了R的值域是由广义矩条件刻画的. (3)主要结果如下。设M是n×k矩阵空间，M是M中矩阵平面空间.从M上的函数到M上的函数的矩阵Radon变换被定义为函数在每个矩阵平面上的积分。然后将矩阵Radon变换的值域刻画为由相应的Cartan运动群产生的广义Pfedian型算子的核。