Zonal spherical harmonics

rdf:langString Zonal spherical harmonics

rdf:langString 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 where Pℓ is a Legendre polynomial of degree ℓ. The general zonal spherical harmonic of degree ℓ is denoted by , 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 In n-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let x be a point on the (n−1)-sphere. Define to be the dual representation of the linear functional in the finite-dimensional Hilbert space Hℓ of spherical harmonics of degree ℓ. In other words, the following reproducing property holds:for all Y ∈ Hℓ. The integral is taken with respect to the invariant probability measure.