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I am looking for good and elementary references on hyperbolic harmonics (which form an orthonormal basis spanning the space of functions on the unit pseudo-sphere).

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brief and elementary:

François Labourie, A short and dirty introduction to hyperbolic surfaces (see chapter 6 on harmonic functions)

more advanced:

Manfred Stoll, Harmonic function theory on real hyperbolic space

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I don't see the answer here. As I understood the question, we need to find a solutions $u$ of the hyperbolic laplacian $\Delta_{h}u=0$ that are harmonic homogenous polynomials such that every function that is solution of the hyperbolic laplacian can be represented as a sum of these hyperbolic harmonic homogenous polynomials. Every hyperbolic harmonic function can be represented as some combination of ordinary spherical harmonics, but what about hyperbolic spherical harmonics, that are solutions of the hyperbolic laplacian and also homogenous polynomials? – Alem Feb 2 '14 at 11:17

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