The invertibility of rotation invariant Radon transforms

Let R_μ denote the Radon transform on Rⁿ that integrates a function over hyperplanes in given smooth positive measures μ depending on the hyperplane. We characterize the measures μ for which R_μ is rotation invariant. We prove rotation invariant transforms are all one-to-one and hence invertible on the domain of square integrable functions of compact support, L₀²(Rⁿ). We prove the hole theorem: if f?L₀²Rⁿ and R_μf = 0 for hyperplanes not intersecting a ball centered at the origin, then f is zero outside of that ball. Using the theory of Fourier integral operators, we extend these results to the domain of distributions of compact support on Rⁿ. Our results prove invertibility for a mathematical model of positron emission tomography and imply a hole theorem for the constantly attenuated Radon transform as well as invertibility for other Radon transforms.