A Gaussian Radon transform for Banach spaces

Abstract

We develop a Radon transform on Banach spaces using Gaussian measure and prove that if a bounded continuous function on a separable Banach space has zero Gaussian integral over all hyperplanes outside a closed bounded convex set in the Hilbert space corresponding to the Gaussian measure then the function is zero outside this set.

The Gaussian Radon transform and machine learning

“…The Gaussian Radon transform generalizes this to infinite dimensions and works with Gaussian measures (instead of Lebesgue measure, for which there is no useful infinite dimensional analog); the motivation for studying this transform comes from the task of reconstructing a random variable from its conditional expectations. We have developed this transform in the setting of abstract Wiener spaces in our work [11] (earlier works, in other frameworks, include [2,3,17]).…”

An Inversion Formula for the Gaussian Radon Transform for Banach Spaces

“…Suppose P = au + u ⊥ is a hyperplane in H, where a ∈ R and u is a unit vector. As shown in [7], if ·, u is continuous on H with respect to the B-norm | · |, then the B-closure P of P in B is a hyperplane in B and, in fact, every hyperplane in B is of this form. However if ·, u is not continuous with respect to | · |, then P = B in B.…”

“…Usually the map in 1.5 is simply denoted h → h, but some measure-theoretic technicalities arising in Section 3 will require us to be a little careful about the true quotient space structure of L 2 -spaces. Although largely self-contained, this work is based on the results in [7], where an infinite-dimensional version of the Radon transform was developed for Banach spaces in the abstract Wiener space setting. In the absence of a useful version of Lebesgue measure on infinite-dimensional spaces, we constructed Gaussian measures on B which are concentrated on hyperplanes and, more generally, on B-closures of closed affine subspaces of H. This result and some needed consequences are presented in Section 2.…”

Limiting Means for Spherical Slices

“…This paper is the fourth in a series of papers. The first [4] develops the Gaussian Radon transform for Banach spaces, where a support theorem was established. The second [11] establishes the result for hyperplanes and the third [10] proves the result for the case of affine planes.…”