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 )=P_{\ell }(\cos \theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22c7c4c135c5f3a50f853f4fca606f600acdc25b)
where
Pℓ is a
Legendre polynomial of degree
ℓ. The general zonal spherical harmonic of degree ℓ is denoted by
![{\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca923861c97a24feb4f33eefc628918151c4e30a)
, 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
In n-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let x be a point on the (n−1)-sphere. Define
to be the dual representation of the linear functional
![{\displaystyle P\mapsto P(\mathbf {x} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd8672293308e1167f09446d0da66b4fa4b957a)
in the finite-dimensional
Hilbert space Hℓ of spherical harmonics of degree ℓ. 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfcbd58d3bf8f61e2b962313ad7d11d7e1baaf02)
for all
Y ∈ Hℓ. The integral is taken with respect to the invariant probability measure.
Relationship with harmonic potentials[edit]
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} ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82d7ea8a7f521e07b5d2edab6b32a159a996e2b2)
where
![{\displaystyle \omega _{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/464bd9d3ab2410f8d3b6a3a31685bcf128699dc4)
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} |)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b504154e78b6e6c5039d43a69c654f203e4477ce)
where
x,y ∈ Rn and the constants
cn,k are given by
![{\displaystyle c_{n,k}={\frac {1}{\omega _{n-1}}}{\frac {2k+n-2}{(n-2)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f046726aa0138893b06f1b3df90ae19fa3d13c4b)
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} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8687f8d0c97aa43a7352b1dc6d23f527903002)
where
cn,ℓ are the constants above and
![{\displaystyle C_{\ell }^{(\alpha )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/916f472d27e0b5e97f28266a81059c5133452d98)
is the ultraspherical polynomial of degree ℓ.
Properties[edit]
- The zonal spherical harmonics are rotationally invariant, meaning that
![{\displaystyle Z_{R\mathbf {x} }^{(\ell )}(R\mathbf {y} )=Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb4479e86d4fea8289bb33f758cd65f0f3768cf6)
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, ..., Yd is an orthonormal basis of Hℓ, then
![{\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )=\sum _{k=1}^{d}Y_{k}(\mathbf {x} ){\overline {Y_{k}(\mathbf {y} )}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c1e547351c05881c6a338da959af9c8375967a)
- Evaluating at x = y gives
![{\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {x} )=\omega _{n-1}^{-1}\dim \mathbf {H} _{\ell }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4548c054efb1f4a7490c2e600da81a8c7356ff44)
References[edit]