A fast transform for spherical harmonics

Spherical harmonics arise on the sphere S² in the same way that the (Fourier) exponential functions {e^ik}k 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(N^5/2 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.