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 where is the normalized Legendre polynomial of degree , where x is a point on the sphere representing the fixed axis, and y

In n-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let x be a point on the (n to be the dual representation in the finite-dimensional Hilbert space with respect to the uniform measure. In other words, we have a reproducing kernel.

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 is the surface area of the (n-1)-dimensional sphere. They are also related to the Newton kernel where and the constants

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 .

Properties

• for every orthogonal transformation R. Conversely, any function on that is a spherical harmonic in y for each fixed x, and that satisfies this invariance property, is a constant multiple of the degree zonal harmonic.

• If Y1, ..., Yd is an orthonormal basis of

• Evaluating at

References

• .