A support theorem and inversion formulas for the Gauss-Radon transform.

The Gauss-Radon transform is a weighted Radon transform using the Gaussian kernel. First we develop a Gaussian measure on a hyperplane in Rn , then we use this measure to develop the Gauss-Radon transform. We construct some examples of Gauss-Radon transforms, including the transforms of polynomial functions and exponential functions. We then develop a Support Theorem for the Gauss-Radon transform similar to a previous theorem for the Radon transform. We then derive three explicit inversion formulas for the Gauss-Radon transform using the relationships between it and the Laplace transform, the Fourier transform, and Segal-Bargmann transform, respectively. Finally, we explore the relationship between Gauss-Radon transforms and conditional expectations and use this to recast several results in a probabilistic setting.