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Zonal spherical harmonics

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Source updated March 5, 2025

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 $Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )$ , 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 $Z_{\mathbf {x} }^{(\ell )}$ 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: $Y(\mathbf {x} )=\int _{S^{n-1}}Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )Y(\mathbf {y} )\,d\Omega (y),\quad \forall Y\in {\mathcal {H}}_{\ell }$ where $\Omega$ is the uniform measure on $\mathbb {S} ^{n-1}$ .

01Relationship 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, ${\frac {1}{\omega _{n-1}}}{\frac {1-r^{2}}{|\mathbf {x} -r\mathbf {y} |^{n}}}=\sum _{k=0}^{\infty }r^{k}Z_{\mathbf {x} }^{(k)}(\mathbf {y} ),$ 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 α = (n−2)/2, then $Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )={\frac {n+2\ell -2}{n-2}}C_{\ell }^{(\alpha )}(\mathbf {x} \cdot \mathbf {y} )$ where $c_{n,\ell }$ are the constants above and $C_{\ell }^{(\alpha )}$ is the ultraspherical polynomial of degree $\ell$ . The 2-dimensional case $Z^{(\ell )}(\theta ,\phi )={\frac {2\ell +1}{4\pi }}P_{\ell }(\cos \theta )$ is a special case of that, since the Legendre polynomials are the special case of the ultraspherical polynomial when $\alpha =1/2$ .

02Properties

The zonal spherical harmonics are rotationally invariant, meaning that $Z_{R\mathbf {x} }^{(\ell )}(R\mathbf {y} )=Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )$ for every orthogonal transformation R. Conversely, any function f(x,y) on Sⁿ⁻¹×Sⁿ⁻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 H_ℓ, then $Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )=\sum _{k=1}^{d}Y_{k}(\mathbf {x} ){\overline {Y_{k}(\mathbf {y} )}}.$
Evaluating at x = y gives $Z_{\mathbf {x} }^{(\ell )}(\mathbf {x} )=\omega _{n-1}^{-1}\dim \mathbf {H} _{\ell }.$

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Sources and credits

This article is adapted from the Wikipedia article “Zonal spherical harmonics”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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