Friday, October 2, 2015

Spherical Harmonic Polar and Equatorial Asymptotics

Spherical Harmonics are eigenfunctions of the Laplace-Beltrami Operator in spherical coordinates. As such they are similar to the Laplace-Beltrami eigenfunctions in Cartesian coordinates (sine/cosine), or Polar coordinates (Bessel functions). The spherical coordinate system can be asymptotically understood in terms of these two 'flat' coordinate systems. Correspondingly, the shape of the lateral part of the spherical harmonics (the associated Legendre functions) behaves like the 'flat' Eigenfunctions, as shown in the two Figures below:
From this perspective, many apparently particular features of the spherical harmonics transform, such as the rotations/Wigner symbols, response to windowing and others have simple analogues in the classical (2D) Fourier transform. Of course, a major difference to flat space are the particular periodic boundary conditions of the spherical coordinate system that induce some of the special spherical properties (e.g. antipodal symmetry). Asymptotics are important because they allow us to understand spherical harmonics operations on the sphere intuitively in terms of 'flat' space operations. I have used the simplest asymptotics in this post, which treat the equator and the poles as perfectly flat regions. More accurate, higher order, approximations cover a larger region of the sphere but are more difficult to understand in terms of the flat Fourier transform properties.

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