The Gaussian Radon Transform for Banach Spaces

• Main Author: Holmes, Irina
• Other Authors: Trahan, Jerry
• Format: Others
• Language: en
• Published: LSU 2014
• Subjects: Mathematics
• Online Access: http://etd.lsu.edu/docs/available/etd-06272014-122519/

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 different directions. A generalization of this transform to infinite-dimensional spaces has the potential to allow one to obtain a function defined on an infinite-dimensional space from its conditional expectations. We work within a standard framework in infinite-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.