# Zonal spherical harmonics

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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

${\displaystyle Z^{(\ell )}(\theta ,\phi )=P_{\ell }(\cos \theta )}$

where P is a Legendre polynomial of degree ℓ. The general zonal spherical harmonic of degree ℓ is denoted by ${\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )}$, 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 of spherical harmonics of degree ℓ. In other words, the following reproducing property holds:

${\displaystyle Y(\mathbf {x} )=\int _{S^{n-1}}Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )Y(\mathbf {y} )\,d\Omega (y)}$

for all Y ∈ H. The integral is taken with respect to the invariant probability measure.

## 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 cn,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 cn,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 ƒ(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.
${\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )=\sum _{k=1}^{d}Y_{k}(\mathbf {x} ){\overline {Y_{k}(\mathbf {y} )}}.}$
• 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.