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Zonal spherical harmonics

In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spherical functions are a broad extension of the notion of zonal spherical harmonics to allow for a more general symmetry group.

On the two-dimensional sphere, the unique zonal spherical harmonic of degree ℓ invariant under rotations fixing the north pole is represented in spherical coordinates by Z^{(\ell)}(\theta,\phi) = \frac{2\ell + 1}{4 \pi} P_\ell(\cos\theta) where P**ℓ is the normalized Legendre polynomial of degree ℓ, P_\ell(1) = 1. The generic zonal spherical harmonic of degree ℓ is denoted by Z^{(\ell)}_{\mathbf{x}}(\mathbf{y}), where x is a point on the sphere representing the fixed axis, and y is the variable of the function. This can be obtained by rotation of the basic zonal harmonic Z^{(\ell)}(\theta,\phi).

In n-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let x be a point on the (n−1)-sphere. Define Z^{(\ell)}{\mathbf{x}} to be the dual representation of the linear functional P\mapsto P(\mathbf{x}) in the finite-dimensional Hilbert space \mathcal H\ell of spherical harmonics of degree \ell with respect to the uniform measure on the sphere \mathbb{S}^{n-1} . In other words, we have a reproducing kernel:Y(\mathbf{x}) = \int_{S^{n-1}} Z^{(\ell)}{\mathbf{x}}(\mathbf{y})Y(\mathbf{y}),d\Omega(y), \quad \forall Y \in \mathcal H\ell where \Omega is the uniform measure on \mathbb{S}^{n-1} .

Relationship with harmonic potentials

The zonal harmonics appear naturally as coefficients of the Poisson kernel for the unit ball in Rn: for x and y unit vectors, \frac{1}{\omega_{n-1}}\frac{1-r^2}{|\mathbf{x}-r\mathbf{y}|^n} = \sum_{k=0}^\infty r^k Z^{(k)}{\mathbf{x}}(\mathbf{y}), where \omega{n-1} is the surface area of the (n-1)-dimensional sphere. They are also related to the Newton kernel via \frac{1}{|\mathbf{x}-\mathbf{y}|^{n-2}} = \sum_{k=0}^\infty c_{n,k} \frac{|\mathbf{x}|^k}{|\mathbf{y}|^{n+k-2}}Z_{\mathbf{x}/|\mathbf{x}|}^{(k)}(\mathbf{y}/|\mathbf{y}|) where x,y ∈ Rn and the constants c**n,k are given by c_{n,k} = \frac{1}{\omega_{n-1}}\frac{2k+n-2}{(n-2)}.

The coefficients of the Taylor series of the Newton kernel (with suitable normalization) are precisely the ultraspherical polynomials. Thus, the zonal spherical harmonics can be expressed as follows. If , then Z^{(\ell)}{\mathbf{x}}(\mathbf{y}) = \frac{n+2\ell-2}{n-2}C\ell^{(\alpha)}(\mathbf{x}\cdot\mathbf{y}) where c_{n, \ell} are the constants above and C_\ell^{(\alpha)} is the ultraspherical polynomial of degree \ell. The 2-dimensional caseZ^{(\ell)}(\theta,\phi) = \frac{2\ell + 1}{4 \pi} P_\ell(\cos\theta)is a special case of that, since the Legendre polynomials are the special case of the ultraspherical polynomial when \alpha = 1/2.

Properties

  • The zonal spherical harmonics are rotationally invariant, meaning that Z^{(\ell)}{R\mathbf{x}}(R\mathbf{y}) = Z^{(\ell)}{\mathbf{x}}(\mathbf{y}) for every orthogonal transformation R. Conversely, any function f(x,y) on S**n−1×S**n−1 that is a spherical harmonic in y for each fixed x, and that satisfies this invariance property, is a constant multiple of the degree ℓ zonal harmonic.
  • If Y1, ..., Y**d is an orthonormal basis of Hℓ, then Z^{(\ell)}{\mathbf{x}}(\mathbf{y}) = \sum{k=1}^d Y_k(\mathbf{x})\overline{Y_k(\mathbf{y})}.
  • Evaluating at gives Z^{(\ell)}{\mathbf{x}}(\mathbf{x}) = \omega{n-1}^{-1} \dim \mathbf{H}_\ell.

References

  • .
Wikipedia Source

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rotational-symmetry special-hypergeometric-functions
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