Horospherical Cauchy Transform on Some Pseudo-Hyperbolic Spaces

TL;DR

This paper explores the horospherical Cauchy transform on hyperboloids, demonstrating how its inversion relates to classical Radon inversion and highlighting the connection between harmonic analysis on symmetric spaces and flat models.

Contribution

It introduces a Cauchy-modified horospherical transform and shows its inversion formulas can be derived from Radon inversion, linking harmonic analysis on symmetric spaces to flat models.

Findings

Inversion formulas derived from Radon inversion

Connection between harmonic analysis on symmetric spaces and flat models

Use of Cauchy modification in horospherical transform

Abstract

We consider the horospherical transform and its inversion in 3 examples of hyperboloids. We want to illustrate via these examples the fact that the horospherical inversion formulas can be directly extracted from the classical Radon inversion formula. In a more broad context, this possibility reflects the fact that the harmonic analysis on symmetric spaces (Riemannian as well as pseudo-Riemannian ones) is equivalent (homologous), up to the Abelian Fourier transform, to the similar problem in the flat model. On the technical level it is important that we work not with the usual horospherical transform, but with its Cauchy modification.