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
( θ , ϕ ) =
( cos θ )
where Pℓ is the normalized Legendre polynomial of degree ℓ,
The generic zonal spherical harmonic of degree ℓ is denoted by
, 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
( θ , ϕ ) .
In n-dimensional Euclidean space, zonal spherical harmonics are defined as follows.
Let x be a point on the (n−1)-sphere.
to be the dual representation of the linear functional
in the finite-dimensional Hilbert space Hℓ of spherical harmonics of degree ℓ with respect to the Haar measure on the sphere
with total mass
(see Unit sphere).
In other words, the following reproducing property holds:
is the Haar measure from above.
The zonal harmonics appear naturally as coefficients of the Poisson kernel for the unit ball in Rn: for x and y unit vectors,
ω
ω
is the surface area of the (n-1)-dimensional sphere.
They are also related to the Newton kernel via
where x,y ∈ Rn and the constants cn,k are given by
ω
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
where cn,ℓ are the constants above and
is the ultraspherical polynomial of degree ℓ.