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

在本研究项目中，我们研究了以下(1)、(2)和(3)。(1)仿射Grassmann流形上的对偶Radon变换。(2)仿射Grassmann流形上Radon变换的矩条件和支持定理。(3)矩阵Radon变换的范围表征。(1)主要结果如下：设G（d,n）为n维欧几里德空间中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。然后用普氏方程组来描述Radon变换R的取值范围。(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，我们证明了R的范围是由广义矩条件表征的。(3)主要结果如下：设M为n×k矩阵空间，Ξ为M中的矩阵平面空间。从M上的函数到Ξ上的函数的矩阵Radon变换定义为每个矩阵平面上的函数的积分。然后将矩阵radon变换的范围表征为对应Cartan运动群产生的广义Pfaffian型算子的核。

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.