Inversions for the Hua-Radon and the polarized Hua-Radon transform

TL;DR

This paper derives inversion formulas for the Hua-Radon and polarized Hua-Radon transforms, which are integral projections on holomorphic functions in the Lie sphere, using kernel integration and spherical monogenic decompositions.

Contribution

It introduces explicit inversion formulas for these transforms, expanding the theoretical understanding of their structure and applications.

Findings

Derived inversion formulas for Hua-Radon and polarized Hua-Radon transforms

Expressed transforms as integral kernels involving zonal spherical monogenics

Utilized Almansi decomposition and reproducing properties for inversion

Abstract

The Hua-Radon and polarized Hua-Radon transform are two orthogonal projections defined on holomorphic functions in the Lie sphere. Both transformations can be written as integral transforms with respect to a suitable reproducing kernel. Integrating both kernels over a Stiefel manifold yields a linear combination of zonal spherical monogenics. Using an Almansi type decomposition of holomorphic functions and reproducing properties of the zonal monogenics, we obtain an inversion formula for both the Hua-Radon and the polarized Hua-Radon transform.