I study a set of points $r_j$ in the hyperbolic plane, let's say in the upper half plane $\{ z\in\mathbb{C} | \Im(z)>0\}$, and I try to define a structure factor in the same spirit as we would define it in crystallography (the Fourier transform of a forest of Dirac deltas): $$S(\mathbf{q})=\sum_{j,k} e^{i \mathbf{q\cdot (r_j-r_k)}}$$
I would like to know what is the scalar product in the case of vectors of the hyperbolic plane so that it would make sense in the complex exponential to define this kind of Fourier transform. Otherwise, I would be interested in any other approach to define the Fourier transform in the hyperbolic geometry.
Here is an attempt based of the analogy with the Euclidian geometry: Since angles are the same in the Euclidian plane and in the upper half plane, we could use u.v=|u||v| cos(u^v)
where u$=(u_1,u_2)$ is a hyperbolic segment in the upper half plane,
|u| is the hyperbolic distance $d(u_1,u_2)$
and u^v is the angle between the directions of u and v.
The value of u^v remains however unknown to me.