We synthesize the double Fourier sphere method and
low rank function techniques to develop a collection of
fast numerical algorithms for computing with functions based
on the fast Fourier transform.  Furthermore, by imposing certain
partial regularity conditions on the solutions of PDEs we derive
optimal complexity and stable spectral methods for long-time
simulations of the Navier-Stokes equations on the disk, spiral waves
on the surface of the sphere, and geophysical flows in the solid
sphere.