Parole chiave

• hypercomplex analysis
• Radon transform
• Dirac delta

Abstract In this talk, we approach the problem of inverting the Radon transform in superspace from two different perspectives. The first one relies on the decomposition into plane waves of the super Dirac Delta distribution, provided that the superdimension is not odd and negative. Such a decomposition is obtained by adopting the point of view of hyperfunctions, namely by using the fact that the Dirac delta is a suitable boundary value of the super Cauchy kernel. In the cases of negative and even superdimension, the obtained formulas no longer resemble the structure of the classical plane wave decompositions in m real dimensions. In turn, the explicit inversion formulas obtained for the super Radon transform in these cases show important differences with the classical case.

On the other hand, we show how to invert the super Radon transform using the classical approach, i.e. by composing the dual Radon transform with a certain power of the super Laplace operator. This approach yields a unified inversion formula that is valid for all possible integer values of the superdimension. The proof of this result comes along with the study of fractional powers of the super Laplacian, their fundamental solutions, and the plane wave decompositions of super Riesz kernels. This talk is based on joint work with Irene Sabadini and Frank Sommen.