On the Bargmann-Radon transform in the monogenic setting

TL;DR

This paper introduces a new Bargmann-Radon transform in the monogenic setting, providing integral representations, characterization formulas, and inversion methods within the real monogenic Bargmann module.

Contribution

It defines and analyzes the Bargmann-Radon transform in the monogenic context, including integral form, characterization, and inversion formulas, extending previous Radon transform concepts.

Findings

Derived an integral form of the Bargmann-Radon transform.

Provided a characterization formula via complex extension and nullcone restriction.

Established an inversion formula for the transform.

Abstract

In this paper we introduce and study a Bargmann-Radon transform on the real monogenic Bargmann module. This transform is defined as the projection of the real Bargmann module on the closed submodule of monogenic functions spanned by the monogenic plane waves. We prove that this projection can be written in integral form in terms the so-called Bargmann-Radon kernel. Moreover, we have a characterization formula for the Bargmann-Radon transform of a function in the real Bargmann module in terms of its complex extension and then its restriction to the nullcone in $\mathbb C^m$. We also show that the formula holds for the Szeg\H{o}-Radon transform that we introduced in [4] (see list of references). Finally, we define the dual transform and we provide an inversion formula.