  We investigate analytic properties of the double Fourier sphere (DFS) method,
which transforms a function defined on the two-dimensional sphere to a function
defined on the two-dimensional torus. Then the resulting function can be
written as a Fourier series yielding an approximation of the original function.
We show that the DFS method preserves smoothness: it continuously maps
spherical H\"older spaces into the respective spaces on the torus, but it does
not preserve spherical Sobolev spaces in the same manner. Furthermore, we prove
sufficient conditions for the absolute convergence of the resulting series
expansion on the sphere as well as results on the speed of convergence.
