### Fast, exact (but unstable) spin spherical harmonic transforms

#### Abstract

In many applications data are measured or defined on a spherical

manifold; spherical harmonic transforms are then required to access

the frequency content of the data. We derive algorithms to perform

forward and inverse spin spherical harmonic transforms for functions

of arbitrary spin number. These algorithms involve recasting the spin

transform on the two-sphere S^2 as a Fourier transform on the

two-torus T^2. 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^3) for any arbitrary spin number, where L is

the harmonic band-limit of the spin function on the sphere. Numerical

experiments are performed and unfortunately the forward transform is

found to be unstable for band-limits above L~32. The 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.

manifold; spherical harmonic transforms are then required to access

the frequency content of the data. We derive algorithms to perform

forward and inverse spin spherical harmonic transforms for functions

of arbitrary spin number. These algorithms involve recasting the spin

transform on the two-sphere S^2 as a Fourier transform on the

two-torus T^2. 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^3) for any arbitrary spin number, where L is

the harmonic band-limit of the spin function on the sphere. Numerical

experiments are performed and unfortunately the forward transform is

found to be unstable for band-limits above L~32. The 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.

#### Keywords

spherical harmonics; spherical harmonic transform; two-sphere; algorithms

#### Full Text:

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