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:


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



Harmonic analysis on constant curvature hyperbolic spaces of arbitrary dimension
This may be explained in Helgason's *Geometric Analysis on Symmetric Spaces*. Theorem 1.3 on Page 201 lists a general Fourier inversion formula. 


If I am parsing everything right, the function $g(\lambda)$ you are asking about is given in the formula as $|\mathbf{c}(\lambda)|^{-2}$, and where $\mathbf{c}$ is [Harish-Chandra's $\mathbf{c}$-function](https://en.wikipedia.org/wiki/Harish-Chandra%27s_c-function). 


