In the [[mathematics|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 function]]s 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 <math display="block">Z^{(\ell)}(\theta,\phi) = \frac{2\ell + 1}{4 \pi} P_\ell(\cos\theta)</math> where {{math|''P''<sub>''ℓ''</sub>}} is the normalized [[Legendre polynomial]] of degree {{mvar|ℓ}}, <math> P_\ell(1) = 1</math>. The generic zonal spherical harmonic of degree ℓ is denoted by <math>Z^{(\ell)}_{\mathbf{x}}(\mathbf{y})</math>, 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 <math>Z^{(\ell)}(\theta,\phi).</math>
In ''n''-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let '''x''' be a point on the (''n''−1)-sphere. Define <math>Z^{(\ell)}_{\mathbf{x}}</math> to be the [[Riesz representation theorem|dual representation]] of the linear functional <math display="block">P\mapsto P(\mathbf{x})</math> in the finite-dimensional [[Hilbert space]] <math>\mathcal H_\ell</math> of spherical harmonics of degree <math>\ell</math> with respect to the [[Haar measure|uniform measure]] on the sphere <math> \mathbb{S}^{n-1} </math>. In other words, we have a [[reproducing kernel]]:<math display="block">Y(\mathbf{x}) = \int_{S^{n-1}} Z^{(\ell)}_{\mathbf{x}}(\mathbf{y})Y(\mathbf{y})\,d\Omega(y), \quad \forall Y \in \mathcal H_\ell</math> where <math> \Omega </math> is the uniform measure on <math> \mathbb{S}^{n-1} </math>.
==Relationship with harmonic potentials== The zonal harmonics appear naturally as coefficients of the [[Poisson kernel]] for the unit ball in '''R'''<sup>''n''</sup>: for '''x''' and '''y''' unit vectors, <math display="block">\frac{1}{\omega_{n-1}}\frac{1-r^2}{|\mathbf{x}-r\mathbf{y}|^n} = \sum_{k=0}^\infty r^k Z^{(k)}_{\mathbf{x}}(\mathbf{y}),</math> where <math>\omega_{n-1}</math> is the surface area of the (n-1)-dimensional sphere. They are also related to the [[Newton kernel]] via <math display="block">\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}|)</math> where {{math|'''x''','''y''' ∈ '''R'''<sup>''n''</sup>}} and the constants {{math|''c''<sub>''n'',''k''</sub>}} are given by <math display="block">c_{n,k} = \frac{1}{\omega_{n-1}}\frac{2k+n-2}{(n-2)}.</math>
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 {{math|1=''α'' = (''n''−2)/2}}, then <math display="block">Z^{(\ell)}_{\mathbf{x}}(\mathbf{y}) = \frac{n+2\ell-2}{n-2}C_\ell^{(\alpha)}(\mathbf{x}\cdot\mathbf{y})</math> where <math>c_{n, \ell}</math> are the constants above and <math>C_\ell^{(\alpha)}</math> is the ultraspherical polynomial of degree <math>\ell</math>. The 2-dimensional case<math display="block">Z^{(\ell)}(\theta,\phi) = \frac{2\ell + 1}{4 \pi} P_\ell(\cos\theta)</math>is a special case of that, since the Legendre polynomials are the special case of the ultraspherical polynomial when <math>\alpha = 1/2</math>.
==Properties==
*The zonal spherical harmonics are rotationally invariant, meaning that <math display="block">Z^{(\ell)}_{R\mathbf{x}}(R\mathbf{y}) = Z^{(\ell)}_{\mathbf{x}}(\mathbf{y})</math> for every orthogonal transformation ''R''. Conversely, any function {{math|''f''(''x'',''y'')}} on {{math|''S''<sup>''n''−1</sup>×''S''<sup>''n''−1</sup>}} that is a spherical harmonic in ''y'' for each fixed ''x'', and that satisfies this invariance property, is a constant multiple of the degree {{mvar|ℓ}} zonal harmonic. *If ''Y''<sub>1</sub>, ..., ''Y''<sub>''d''</sub> is an [[orthonormal basis]] of {{math|'''H'''<sub>''ℓ''</sub>}}, then <math display="block">Z^{(\ell)}_{\mathbf{x}}(\mathbf{y}) = \sum_{k=1}^d Y_k(\mathbf{x})\overline{Y_k(\mathbf{y})}.</math> *Evaluating at {{math|1='''x''' = '''y'''}} gives <math display="block">Z^{(\ell)}_{\mathbf{x}}(\mathbf{x}) = \omega_{n-1}^{-1} \dim \mathbf{H}_\ell.</math>
==References== * {{citation|last1=Stein|first1=Elias|authorlink1=Elias Stein|first2=Guido|last2=Weiss|authorlink2=Guido Weiss|title=Introduction to Fourier Analysis on Euclidean Spaces|publisher=Princeton University Press|year=1971|isbn=978-0-691-08078-9|location=Princeton, N.J.| url-access=registration|url=https://archive.org/details/introductiontofo0000stei}}.
[[Category:Rotational symmetry]] [[Category:Special hypergeometric functions]]