Fast, exact (but unstable) spin spherical harmonic transforms

We derive algorithms to perform a spin spherical harmonic transform and inverse for functions of arbitrary spin number. These algorithms involve recasting the spin transform on the two-sphere S as a Fourier transform on the two-torus T. Fast Fourier transforms are then used to compute Fourier coefficients, which are related to spherical harmonic coefficients through a linear transform. By recasting the problem as a Fourier transform on the torus we appeal to the usual Shannon sampling theorem to develop spherical harmonic transforms that are theoretically exact for band-limited functions, thereby providing an alternative sampling theorem on the sphere. The computational complexity of our forward and inverse spin spherical harmonic transforms scale as O(L) for any arbitrary spin number, where L is the harmonic band-limit of the spin function on the sphere. The algorithms also apply for functions with arbitrary band-limit and not only powers of two. Numerical experiments are performed and unfortunately the forward transform is found to be unstable for band-limits above L ≃ 32. The source of this instability is due to the poorly conditioned linear system relating Fourier and spherical harmonic coefficients. The inverse transform is expected to be stable, although it is not possible to verify this hypothesis. In ongoing work we are attempting to renormalise or reformulate the algorithms in such a way as to eliminate the numerical stability problem.