Processes of flats induced by higher dimensional processes III

Abstract

Stationary processes of k-flats in $\mathbb{E}$ ^d can be thought of as point processes on the Grassmannian $\mathcal{L}$ _k ^d of k-dimensional subspaces of $\mathbb{E}$ ^d. If such a process is sampled by a (d−k+ j)-dimensional space F, it induces a process of j-flats in F. In this work we will investigate the possibility of determining the original k-process from knowledge of the intensity measures of the induced j-processes. We will see that this is impossible precisely when 1<k<d−1 and j=0,...,2[r/2]−1, where r is the rank of the manifold $\mathcal{L}$ _k ^d. We will show how the problem is equivalent to the study of the kernel of various integral transforms, these will then be investigated using harmonic analysis on Grassmannian manifolds.