On the Bargmann-Radon transform in the monogenic setting

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 Cm. We also show that the formula holds for the Szegő–Radon transform that we introduced in Colombo et al. (2016) and we define the dual transform and we provide an inversion formula. Finally, in Theorem 5.6, we prove an integral formula for the monogenic part of an entire function.