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 $${\displaystyle Z^{(\ell )}(\theta ,\phi )={\frac {2\ell +1}{4\pi }}P_{\ell }(\cos \theta )}$$ where P_ℓ is the normalized Legendre polynomial of degree ℓ, ${\displaystyle P_{\ell }(1)=1}$. The generic zonal spherical harmonic of degree ℓ is denoted by ${\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )}$, 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 ${\displaystyle Z^{(\ell )}(\theta ,\phi ).}$

In n-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let x be a point on the (n−1)-sphere. Define ${\displaystyle Z_{\mathbf {x} }^{(\ell )}}$ to be the dual representation of the linear functional $${\displaystyle P\mapsto P(\mathbf {x} )}$$ in the finite-dimensional Hilbert space H_ℓ of spherical harmonics of degree ℓ with respect to the Haar measure on the sphere ${\displaystyle \mathbb {S} ^{n-1}}$ with total mass ${\displaystyle A_{n-1}}$ (see Unit sphere). In other words, the following reproducing property holds: $${\displaystyle Y(\mathbf {x} )=\int _{S^{n-1}}Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )Y(\mathbf {y} )\,d\Omega (y)}$$ for all Y∈H_ℓ where ${\displaystyle \Omega }$ is the Haar measure from above.

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, $${\displaystyle {\frac {1}{\omega _{n-1}}}{\frac {1-r^{2}}{|\mathbf {x} -r\mathbf {y} |^{n}}}=\sum _{k=0}^{\infty }r^{k}Z_{\mathbf {x} }^{(k)}(\mathbf {y} ),}$$ where ${\displaystyle \omega _{n-1}}$ is the surface area of the (n-1)-dimensional sphere. They are also related to the Newton kernel via $${\displaystyle {\frac {1}{|\mathbf {x} -\mathbf {y} |^{n-2}}}=\sum _{k=0}^{\infty }c_{n,k}{\frac {|\mathbf {x} |^{k}}{|\mathbf {y} |^{n+k-2}}}Z_{\mathbf {x} /|\mathbf {x} |}^{(k)}(\mathbf {y} /|\mathbf {y} |)}$$ where x,y∈Rⁿ and the constants c_n,k are given by $${\displaystyle c_{n,k}={\frac {1}{\omega _{n-1}}}{\frac {2k+n-2}{(n-2)}}.}$$

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 $${\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )={\frac {n+2\ell -2}{n-2}}C_{\ell }^{(\alpha )}(\mathbf {x} \cdot \mathbf {y} )}$$ where c_n,ℓ are the constants above and ${\displaystyle C_{\ell }^{(\alpha )}}$ is the ultraspherical polynomial of degree ℓ.

Properties

• The zonal spherical harmonics are rotationally invariant, meaning that $${\displaystyle Z_{R\mathbf {x} }^{(\ell )}(R\mathbf {y} )=Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )}$$ 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 $${\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )=\sum _{k=1}^{d}Y_{k}(\mathbf {x} ){\overline {Y_{k}(\mathbf {y} )}}.}$$
• Evaluating at x = y gives $${\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {x} )=\omega _{n-1}^{-1}\dim \mathbf {H} _{\ell }.}$$

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 .