A fast transform for spherical harmonics

Abstract

Spherical harmonics arise on the sphere S2 in the same way that the (Fourier) exponential functions {eikθ}k∈ℤ arise on the circle. Spherical harmonic series have many of the same wonderful properties as Fourier series, but have lacked one important thing: a numerically stable fast transform analogous to the Fast Fourier Transform (FFT). Without a fast transform, evaluating (or expanding in) spherical harmonic series on the computer is slow—for large computations probibitively slow. This paper provides a fast transform.

For a grid ofO(N2) points on the sphere, a direct calculation has computational complexityO(N4), but a simple separation of variables and FFT reduce it toO(N3) time. Here we present algorithms with timesO(N5/2 log N) andO(N2(log N)2).

The problem quickly reduces to the fast application of matrices of associated Legendre functions of certain orders. The essential insight is that although these matrices are dense and oscillatory, locally they can be represented efficiently in trigonometric series.

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Communicated by M. Victor Wickerhauser

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Mohlenkamp, M.J. A fast transform for spherical harmonics. The Journal of Fourier Analysis and Applications 5, 159–184 (1999). https://doi.org/10.1007/BF01261607

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Math subject classifications

  • Primary 65T20
  • secondary 42C10
  • 33C55

Keywords and phrases

  • spherical harmonics
  • fast transforms
  • associated Legendre functions