  The double Fourier sphere (DFS) method uses a clever trick to transform a
function defined on the unit sphere to the torus and subsequently approximate
it by a Fourier series, which can be evaluated efficiently via fast Fourier
transforms. Similar approaches have emerged for approximation problems on the
disk, the ball, and the cylinder. In this paper, we introduce a generalized DFS
method applicable to various manifolds, including all the above-mentioned cases
and many more, such as the rotation group. This approach consists in
transforming a function defined on a manifold to the torus of the same
dimension. We show that the Fourier series of the transformed function can be
transferred back to the manifold, where it converges uniformly to the original
function. In particular, we obtain analytic convergence rates in case of
H\"older-continuous functions on the manifold.
