Spherical coordinates

The spherical harmonic transform is the natural extension of the Fourier series to the surface of a sphere, used for functions defined on spherical domains. It expresses a function in terms of spherical harmonics, \(Y_{\ell m}(\theta, \phi)\), where \(\ell\) and \(m\) are integers that index the degree and order of the harmonics, respectively. The spherical harmonics are the angular portion of the solution to Laplace’s equation in spherical coordinates, making this transform useful in solving problems on spherical domains, like those in geophysics or astrophysics.

cf the Hankel transform, which is kind of a radial version of the Fourier transform, characterising dependence upon radius. That is typically used for problems with radial symmetry, such as wave propagation, heat flow, or Laplace’s equation in circular geometries. The Hankel transform of order \(n\) is defined using the Bessel function of the first kind, \(J_n(x)\), as the kernel: \[ \mathcal{H}_n[f(r)](k) = \int_0^\infty f(r) J_n(kr) r \, dr \]

Spherical harmonics handle the angular dependence of functions on a sphere, while the Hankel Transform handles the radial dependence. When solving PDEs in spherical coordinates, it’s common to represent the angular part of the solution using spherical harmonics and the radial part using the Hankel.