Non-Euclidean Fourier Inversion on Super-hyperbolic Space

What is it about?

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.

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