Higher Dimensions
The classical spherical harmonics are defined as functions on the unit sphere S2 inside three-dimensional Euclidean space. Spherical harmonics can be generalized to higher dimensional Euclidean space Rn as follows. Let Pℓ denote the space of homogeneous polynomials of degree ℓ in n variables. That is, a polynomial P is in Pℓ provided that
Let Aℓ denote the subspace of Pℓ consisting of all harmonic polynomials; these are the solid spherical harmonics. Let Hℓ denote the space of functions on the unit sphere
obtained by restriction from Aℓ.
The following properties hold:
- The spaces Hℓ are dense in the set of continuous functions on Sn−1 with respect to the uniform topology, by the Stone-Weierstrass theorem. As a result, they are also dense in the space L2(Sn−1) of square-integrable functions on the sphere.
- For all ƒ ∈ Hℓ, one has
-
- where ΔSn−1 is the Laplace–Beltrami operator on Sn−1. This operator is the analog of the angular part of the Laplacian in three dimensions; to wit, the Laplacian in n dimensions decomposes as
- It follows from the Stokes theorem and the preceding property that the spaces Hℓ are orthogonal with respect to the inner product from L2(Sn−1). That is to say,
-
- for ƒ ∈ Hℓ and g ∈ Hk for k ≠ ℓ.
- Conversely, the spaces Hℓ are precisely the eigenspaces of ΔSn−1. In particular, an application of the spectral theorem to the Riesz potential gives another proof that the spaces Hℓ are pairwise orthogonal and complete in L2(Sn−1).
- Every homogeneous polynomial P ∈ Pℓ can be uniquely written in the form
-
- where Pj ∈ Aj. In particular,
An orthogonal basis of spherical harmonics in higher dimensions can be constructed inductively by the method of separation of variables, by solving the Sturm-Liouville problem for the spherical Laplacian
where φ is the axial coordinate in a spherical coordinate system on Sn−1. The end result of such a procedure is
where the indices satisfy |ℓ1| ≤ ℓ2 ≤ ... ≤ ℓn−1 and the eigenvalue is −ℓn−1(ℓn−1 + n−2). The functions in the product are defined in terms of the Legendre function
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