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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2 Answers
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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
answered Sep 18, 2012 at 11:51
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$\begingroup$ 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? $\endgroup$– AlemCommented Feb 2, 2014 at 11:17
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It seems you are seeking the homogeneous angular eigenfunctions of the hyperbolic Laplacian $\nabla^2_{\mathbb{H}^n} u = 0$ defined on a real Lobachevsky space of dimension $n$. In that case, the following references might be helpful:
- Askold M. Perelomov, Generalized Coherent States and Their Applications (see Chapter 7 for a detailed discussion, and Chapters 5, 6, and 14 for related special cases).
- Audrey Terras, Harmonic Analysis on Symmetric Spaces and Applications, Vol. I (refer to Chapter III for a relevant treatment).
For additional related discussions, consider the following works:
- Isaac Chavel, Eigenvalues in Riemannian Geometry.
- Sirkka-Liisa Eriksson & Terhi Kaarakka, Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion (Adv. Appl. Clifford Algebras, Vol. 30, No. 72, 2020).
- Selberg trace formula.
