The Radon transform between monogenic and generalized slice monogenic functions

Abstract

In Bureš et al. (Elements of quaternionic analysis and Radon transform, 2009), the authors describe a link between holomorphic functions depending on a parameter and monogenic functions defined on ${\mathbb {R}}^{n+1}$ using the Radon and dual Radon transforms. The main aim of this paper is to further develop this approach. In fact, the Radon transform for functions with values in the Clifford algebra ${\mathbb {R}}_n$ is mapping solutions of the generalized Cauchy–Riemann equation, i.e., monogenic functions, to a parametric family of holomorphic functions with values in ${\mathbb {R}}_n$ and, analogously, the dual Radon transform is mapping parametric families of holomorphic functions as above to monogenic functions. The parametric families of holomorphic functions considered in the paper can be viewed as a generalization of the so-called slice monogenic functions. An important part of the problem solved in the paper is to find a suitable definition of the function spaces serving as the domain and the target of both integral transforms.