The Radon transform between monogenic and generalized slice monogenic functions

TL;DR

This paper explores the relationship between monogenic functions and generalized slice monogenic functions through Radon and dual Radon transforms, extending quaternionic analysis to Clifford algebra-valued functions.

Contribution

It develops a framework for the Radon transform connecting monogenic and holomorphic functions in Clifford algebra, generalizing slice monogenic functions.

Findings

Radon transform maps monogenic functions to holomorphic functions in R_n.

Dual Radon transform maps parametric holomorphic functions back to monogenic functions.

Established suitable function spaces for the integral transforms.

Abstract

In [J. Bures, R. Lavicka, V. Soucek, Elements of quaternionic analysis and Radon transform, Textos de Matematica 42, Departamento de Matematica, Universidade de Coimbra, 2009], the authors describe a link between holomorphic functions depending on a parameter and monogenic functions defined on 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 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 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.