  We develop a novel sampling theorem on the sphere and corresponding fast
algorithms by associating the sphere with the torus through a periodic
extension. The fundamental property of any sampling theorem is the number of
samples required to represent a band-limited signal. To represent exactly a
signal on the sphere band-limited at L, all sampling theorems on the sphere
require O(L^2) samples. However, our sampling theorem requires less than half
the number of samples of other equiangular sampling theorems on the sphere and
an asymptotically identical, but smaller, number of samples than the
Gauss-Legendre sampling theorem. The complexity of our algorithms scale as
O(L^3), however, the continual use of fast Fourier transforms reduces the
constant prefactor associated with the asymptotic scaling considerably,
resulting in algorithms that are fast. Furthermore, we do not require any
precomputation and our algorithms apply to both scalar and spin functions on
the sphere without any change in computational complexity or computation time.
We make our implementation of these algorithms available publicly and perform
numerical experiments demonstrating their speed and accuracy up to very high
band-limits. Finally, we highlight the advantages of our sampling theorem in
the context of potential applications, notably in the field of compressive
sampling.
