A fast transform for spherical harmonics

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.