Doctor of Philosophy (PhD)
The classical Radon transform can be thought of as a way to obtain the density of an n-dimensional object from its (n-1)-dimensional sections in diff_x001B_erent directions. A generalization of this transform to infi_x001C_nite-dimensional spaces has the potential to allow one to obtain a function de_x001C_fined on an infi_x001C_nite-dimensional space from its conditional expectations. We work within a standard framework in in_x001C_finite-dimensional analysis, that of abstract Wiener spaces, developed by L. Gross. The main obstacle in infinite dimensions is the absence of a useful version of Lebesgue measure. To overcome this, we work with Gaussian measures. Specifically, we construct Gaussian measures concentrated on closed affine subspaces of infinite-dimensional Banach spaces, and use these measures to define the Gaussian Radon transform. We provide for this transform a disintegration theorem, an inversion procedure and explore possible applications to machine learning.
Document Availability at the Time of Submission
Release the entire work immediately for access worldwide.
Holmes, Irina, "The Gaussian Radon Transform for Banach Spaces" (2014). LSU Doctoral Dissertations. 2040.