The monogenic Hua-Radon transform and its inverse

TL;DR

This paper introduces the monogenic Hua-Radon transform, determines its reproducing kernel, and provides an explicit inversion formula by integrating over the Stiefel manifold, advancing the understanding of holomorphic functions in Lie sphere geometry.

Contribution

It extends previous work by explicitly deriving the reproducing kernel and inversion formula for the monogenic Hua-Radon transform in Lie sphere geometry.

Findings

Reproducing kernel of the monogenic Hua-Radon transform is explicitly determined.

Inversion formula for the transform is obtained via integration over the Stiefel manifold.

The work generalizes previous results on holomorphic functions in Lie sphere context.

Abstract

The monogenic Hua-Radon transform is defined as an orthogonal projection on holomorphic functions in the Lie sphere. Extending the work of Sabadini and Sommen, J. Geom. Anal. 29 (2019), 2709-2737, we determine its reproducing kernel. Integrating this kernel over the Stiefel manifold yields a linear combination of the zonal spherical monogenics. Using the reproducing properties of those monogenics we obtain an inversion for the monogenic Hua-Radon transform.