I am currently looking for a formulation of a Fourier transform on manifolds of constant negative curvature. Specifically, I am looking for generalizations of the two dimensional results on the poincare disk parameterization given in [1].

Using notation $x = (x_1,x_2,x_3,\ldots)$, I have found that the metric of the poincare disk generalizes to a poincare ball , taking the points with $x_1^2+x_2^2+x_3^2+\ldots <1$ (e.g. $x\in B_1$), and using the metric $\frac{d x_1^2+dx_2^2+dx_3^2+\ldots}{1-|x|^2}$. For ease of notation I call the dimension of the poincare ball $d$.

I have already found that (some of) the eigenfunctions of the laplace-beltrami operator are given (from [1]) by functions of the form $e^{-\mu \tanh^{-1}\left(1+\frac{1-|x|^2}{1-x\cdot b}\right)}$, where $b=(b_1,b_2,b_3\ldots)\in S^d$ (e.g. $b_1^2+b_2^2+b_3^2+\ldots = 1$). The eigenvalues of these functions are given by $\mu(\mu-2d+2)$.

I then tried to define a forward fourier transform through $\bar{f}(\lambda,b) = \int_{B_1}f(x)e^{-(-i\lambda+d-1) \tanh^{-1}\left(1+\frac{1-|x|^2}{1-x\cdot b}\right)}$, mirroring the 2-dimensional result. However, I am unable to find a proper inverse transform. Based on symmetry in the formulation, my best assumption at the moment is that the inverse transform is of the form $\int_{\mathbb{R}}\int_{S^d}\bar{f}(\lambda,b)e^{-(i\lambda+d-1) \tanh^{-1}\left(1+\frac{1-|x|^2}{1-x\cdot b}\right)}g(\lambda)dbd\lambda$. Again, due to helgason[1] I already know that for the two dimensional case, $g(\lambda) = \frac{1}{4\pi}\lambda\tanh(\frac{\pi\lambda}{2})$, but I have been unable to find the proper generalization for this in higher dimensions. Concretely, my question is thus: how does $g(\lambda)$ generalize to higher dimensions? Alternatively, if someone could point me to a fourier type transform for some other parameterization of hyperbolic space in $d$ dimensions that would also be immensely useful.

I have so far tried to approach the problem by learning and applying techniques of harmonic analysis. Unfortunately, the introductory books I have found on the subject[2,3] do not seem to deal with the locally compact non-abelian case.

References:

- Sigurdur Helgason, Topics in Harmonic Analysis on Homogeneous Spaces
- Yitzhak Katznelson, An introduction to harmonic analysis
- Gerald B. Folland, A course in abstract harmonic analysis.