# Zonal spherical harmonics

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In the [mathematical](/source/Mathematics) study of [rotational symmetry](/source/Rotational_symmetry), the **zonal spherical harmonics** are special [spherical harmonics](/source/Spherical_harmonics) that are invariant under the rotation through a particular fixed axis. The [zonal spherical functions](/source/Zonal_spherical_function) are a broad extension of the notion of zonal spherical harmonics to allow for a more general [symmetry group](/source/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](/source/Spherical_coordinates) by Z ( ℓ ) ( θ , ϕ ) = 2 ℓ + 1 4 π P ℓ ( cos ⁡ θ ) {\displaystyle Z^{(\ell )}(\theta ,\phi )={\frac {2\ell +1}{4\pi }}P_{\ell }(\cos \theta )} where *P**ℓ* is the normalized [Legendre polynomial](/source/Legendre_polynomial) of degree ℓ, P ℓ ( 1 ) = 1 {\displaystyle P_{\ell }(1)=1} . The generic zonal spherical harmonic of degree ℓ is denoted by Z x ( ℓ ) ( y ) {\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 Z ( ℓ ) ( θ , ϕ ) . {\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 Z x ( ℓ ) {\displaystyle Z_{\mathbf {x} }^{(\ell )}} to be the [dual representation](/source/Riesz_representation_theorem) of the linear functional P ↦ P ( x ) {\displaystyle P\mapsto P(\mathbf {x} )} in the finite-dimensional [Hilbert space](/source/Hilbert_space) H ℓ {\displaystyle {\mathcal {H}}_{\ell }} of spherical harmonics of degree ℓ {\displaystyle \ell } with respect to the [uniform measure](/source/Haar_measure) on the sphere S n − 1 {\displaystyle \mathbb {S} ^{n-1}} . In other words, we have a [reproducing kernel](/source/Reproducing_kernel): Y ( x ) = ∫ S n − 1 Z x ( ℓ ) ( y ) Y ( y ) d Ω ( y ) , ∀ Y ∈ H ℓ {\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 Ω {\displaystyle \Omega } is the uniform measure on S n − 1 {\displaystyle \mathbb {S} ^{n-1}} .

## Relationship with harmonic potentials

The zonal harmonics appear naturally as coefficients of the [Poisson kernel](/source/Poisson_kernel) for the unit ball in **R***n*: for **x** and **y** unit vectors, 1 ω n − 1 1 − r 2 | x − r y | n = ∑ k = 0 ∞ r k Z x ( k ) ( y ) , {\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 ω n − 1 {\displaystyle \omega _{n-1}} is the surface area of the (n-1)-dimensional sphere. They are also related to the [Newton kernel](/source/Newton_kernel) via 1 | x − y | n − 2 = ∑ k = 0 ∞ c n , k | x | k | y | n + k − 2 Z x / | x | ( k ) ( y / | y | ) {\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** ∈ **R***n* and the constants *c**n*,*k* are given by c n , k = 1 ω n − 1 2 k + n − 2 ( n − 2 ) . {\displaystyle c_{n,k}={\frac {1}{\omega _{n-1}}}{\frac {2k+n-2}{(n-2)}}.}

The coefficients of the [Taylor series](/source/Taylor_series) of the Newton kernel (with suitable normalization) are precisely the [ultraspherical polynomials](/source/Ultraspherical_polynomials). Thus, the zonal spherical harmonics can be expressed as follows. If *α* = (*n*−2)/2, then Z x ( ℓ ) ( y ) = n + 2 ℓ − 2 n − 2 C ℓ ( α ) ( x ⋅ y ) {\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 {\displaystyle c_{n,\ell }} are the constants above and C ℓ ( α ) {\displaystyle C_{\ell }^{(\alpha )}} is the ultraspherical polynomial of degree ℓ {\displaystyle \ell } . The 2-dimensional case Z ( ℓ ) ( θ , ϕ ) = 2 ℓ + 1 4 π P ℓ ( cos ⁡ θ ) {\displaystyle Z^{(\ell )}(\theta ,\phi )={\frac {2\ell +1}{4\pi }}P_{\ell }(\cos \theta )} is a special case of that, since the Legendre polynomials are the special case of the ultraspherical polynomial when α = 1 / 2 {\displaystyle \alpha =1/2} .

## Properties

- The zonal spherical harmonics are rotationally invariant, meaning that Z R x ( ℓ ) ( R y ) = Z x ( ℓ ) ( y ) {\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 *S**n*−1×*S**n*−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 *Y*1, ..., *Y**d* is an [orthonormal basis](/source/Orthonormal_basis) of **H***ℓ*, then Z x ( ℓ ) ( y ) = ∑ k = 1 d Y k ( x ) Y k ( y ) ¯ . {\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 Z x ( ℓ ) ( x ) = ω n − 1 − 1 dim ⁡ H ℓ . {\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {x} )=\omega _{n-1}^{-1}\dim \mathbf {H} _{\ell }.}

## References

- [Stein, Elias](/source/Elias_Stein); [Weiss, Guido](/source/Guido_Weiss) (1971), [*Introduction to Fourier Analysis on Euclidean Spaces*](https://archive.org/details/introductiontofo0000stei), Princeton, N.J.: Princeton University Press, [ISBN](/source/ISBN_(identifier)) [978-0-691-08078-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-08078-9).

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Adapted from the Wikipedia article [Zonal spherical harmonics](https://en.wikipedia.org/wiki/Zonal_spherical_harmonics) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Zonal_spherical_harmonics?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.