The Gaussian Radon Transform for Banach Spaces

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 di erent directions. A generalization of this transform to in nite-dimensional spaces has the potential to allow one to obtain a function de ned on an in nite-dimensional space from its conditional expectations. We work within a standard framework in in nite-dimensional analysis, that of abstract Wiener spaces, developed by L. Gross. The main obstacle in in nite dimensions is the absence of a useful version of Lebesgue measure. To overcome this, we work with Gaussian measures. Speci cally, we construct Gaussian measures concentrated on closed a ne subspaces of in nitedimensional Banach spaces, and use these measures to de ne the Gaussian Radon transform. We provide for this transform a disintegration theorem, an inversion procedure and explore possible applications to machine learning.