On the Radon transform and the Dirac delta distribution in superspace

TL;DR

This paper develops a plane wave decomposition for the delta distribution in superspace, enabling explicit inversion formulas for the super Radon transform across various superdimensions, and explores fractional powers of the super Laplacian.

Contribution

It introduces a general Radon inversion formula valid for all integer superdimensions and studies fractional super Laplacians and super Riesz kernels.

Findings

Explicit inversion formulas for super Radon transform

Plane wave decomposition for delta distribution in superspace

General Radon inversion formula for all superdimensions

Abstract

In this manuscript, we obtain a plane wave decomposition for the delta distribution in superspace, provided that the superdimension is not odd and negative. This decomposition allows for explicit inversion formulas for the super Radon transform in these cases. Moreover, we prove a more general Radon inversion formula 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.