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 $$Z^{(\ell )}(\theta ,\varphi )=\frac{2\ell +1}{4\pi }{P}_{\ell }(\cos \theta )$$ 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 $Z^{(\ell )}(\theta ,\varphi ).$

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 ${{\mathscr{H}}}_{\ell }$ 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:$$Y(\mathbf{𝐱})=\underset{S^{n-1}}{\int }{Z}_{\mathbf{𝐱}}^{(\ell )}(\mathbf{𝐲})Y(\mathbf{𝐲})d\mathrm{\Omega }(y),\mathrm{\forall }Y\in {{\mathscr{H}}}_{\ell }$$ where $\mathrm{\Omega }$ 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, $$\frac{1}{{\omega }_{n-1}}\frac{1-r^2}{|\mathbf{𝐱}-r\mathbf{𝐲}{|}^n}=\sum _{k=0}^{\mathrm{\infty }}{r}^k{Z}_{\mathbf{𝐱}}^{(k)}(\mathbf{𝐲}),$$ where ${\omega }_{n-1}$ is the surface area of the (n-1)-dimensional sphere. They are also related to the Newton kernel via $$\frac{1}{|\mathbf{𝐱}-\mathbf{𝐲}{|}^{n-2}}=\sum _{k=0}^{\mathrm{\infty }}{c}_{n,k}\frac{|\mathbf{𝐱}{|}^k}{|\mathbf{𝐲}{|}^{n+k-2}}{Z}_{\mathbf{𝐱}/|\mathbf{𝐱}|}^{(k)}(\mathbf{𝐲}/|\mathbf{𝐲}|)$$ 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$$Z^{(\ell )}(\theta ,\varphi )=\frac{2\ell +1}{4\pi }{P}_{\ell }(\cos \theta )$$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 $$Z_{R\mathbf{𝐱}}^{(\ell )}(R\mathbf{𝐲})=Z_{\mathbf{𝐱}}^{(\ell )}(\mathbf{𝐲})$$ 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 $$Z_{\mathbf{𝐱}}^{(\ell )}(\mathbf{𝐱})={\omega }_{n-1}^{-1}\dim {\mathbf{𝐇}}_{\ell }.$$

References

• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9, https://archive.org/details/introductiontofo0000stei .