Skip to main content
Article thumbnail
Location of Repository

A fast transform for spherical harmonics

By Martin J. Mohlenkamp


Acknowledgements and Notes. I would like to thank my thesis advisor, R.R. Coifman, for his help and guidance. Spherical Harmonics arise on the sphere S 2 in the same way that the (Fourier) exponential functions {e ikθ}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. Without a fast transform, evaluating (or expanding in) Spherical Harmonic series on the computer is slow—for large computations prohibitively slow. This paper provides a fast transform. For a grid of O(N 2) points on the sphere, a direct calculation has computational complexity O(N 4), but a simple separation of variables and Fast Fourier Transform reduce it to O(N 3) time. Here we present algorithms with times O(N 5/2 log N) and O(N 2 (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.

Topics: Fast Transforms, Associated Legendre Functions
Year: 1999
OAI identifier: oai:CiteSeerX.psu:
Provided by: CiteSeerX
Download PDF:
Sorry, we are unable to provide the full text but you may find it at the following location(s):
  • (external link)
  • (external link)
  • Suggested articles

    To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.