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

Z ( ℓ ) ( θ , ϕ ) = P ℓ ( cos ⁡ θ ) {\displaystyle Z^{(\ell )}(\theta ,\phi )=P_{\ell }(\cos \theta )}
where Pℓ is a Legendre polynomial of degree ℓ. The general 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 of the linear functional

P ↦ P ( x ) {\displaystyle P\mapsto P(\mathbf {x} )}
in the finite-dimensional Hilbert space Hℓ of spherical harmonics of degree ℓ. In other words, the following reproducing property holds:
Y ( x ) = ∫ S n − 1 Z x ( ℓ ) ( y ) Y ( y ) d Ω ( y ) {\displaystyle Y(\mathbf {x} )=\int _{S^{n-1}}Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )Y(\mathbf {y} )\,d\Omega (y)}
for all Y ∈ Hℓ. 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,

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 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 ∈ Rn and the constants cn,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 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

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 cn,ℓ are the constants above and C ℓ ( α ) {\displaystyle C_{\ell }^{(\alpha )}} is the ultraspherical polynomial of degree ℓ.

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 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
    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; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9.