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

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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.