Non-Euclidean Fourier inversion on super-hyperbolic space

TL;DR

This paper develops a non-Euclidean Fourier transform for super-hyperbolic spaces, providing an inversion formula that reveals spectral components, with detailed analysis of superfunctions and boundary behaviors.

Contribution

It introduces the first non-Euclidean Helgason--Fourier transform for super-hyperbolic spaces and establishes an inversion formula with residue contributions, advancing harmonic analysis in supergeometry.

Findings

Inversion formula with residue contributions at poles

Detailed analysis of spherical superfunctions

Rigorous boundary term estimates in polar coordinates

Abstract

For the super-hyperbolic space in any dimension, we introduce the non-Euclidean Helgason--Fourier transform. We prove an inversion formula exhibiting residue contributions at the poles of the Harish-Chandra c-function, signalling discrete parts in the spectrum. The proof is based on a detailed study of the spherical superfunctions, using recursion relations and localization techniques to normalize them precisely, careful estimates of their derivatives, and a rigorous analysis of the boundary terms appearing in the polar coordinate expression of the invariant integral