Views on the Radon Transform

Abstract: I will recall and introduce some of the many existing Radon transforms, focusing in particular on the setup of $G$-dual pairs $(X,\Xi)$ introduced by Helgason more than fifty years ago, where $G$ is a locally compact group that acts transitively both on $X$ and $\Xi$. I will then present some results obtained in collaboration with G. S. Alberti, F. Bartolucci, E. De Vito, M. Monti and F. Odone which bring into play (square integrable) representations. If the functions to be analyzed live on $X$ and the quasi regular representation of $G$ on $L^2(X)$ and $L^2(\Xi)$ are square integrable, then it is possible to write a nice inversion formula for the Radon transform associated to the families of submanifolds of $X$ that are prescribed by the object $\Xi$ which is dual to $X$. This formula hinges on a unitarization of the Radon transform that may be proved in a rather general setup if the quasi regular representations of $G$ on $L^2(X)$ and $L^2(\Xi)$ are irreducible, and on an intertwining property of the Radon transform. The former result is inspired by work of Helgason. Some examples are discussed, mostly the guiding case related to shearlets that points in the direction of possible practical inversion techniques.

classical analysis and ODEsfunctional analysisrepresentation theoryspectral theory

Audience: researchers in the topic

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