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 ${\displaystyle {\mathcal {H}}_{\ell }}$ of spherical harmonics of degree ${\displaystyle \ell }$ with respect to the uniform measure on the sphere ${\displaystyle \mathbb {S} ^{n-1}}$. In other words, we have a reproducing kernel:$${\displaystyle Y(\mathbf {x} )=\int _{S^{n-1}}Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )Y(\mathbf {y} )\,d\Omega (y),\quad \forall Y\in {\mathcal {H}}_{\ell }}$$ where ${\displaystyle \Omega }$ is the uniform measure on ${\displaystyle \mathbb {S} ^{n-1}}$.