On the Role of the Double Fourier Sphere Method in Fast Algorithms on SO(3)

We analyze the Double Fourier Sphere (DFS) method on the rotation group $\mathcal{S}\mathcal{O}(3)$ in the frequency domain and demonstrate its central role in fast algorithms. Fast Fourier algorithms on $\mathcal{S}\mathcal{O}(3)$ are commonly formulated as a Wigner transform - mapping harmonic to Fourier coefficients - followed by a Fourier transform. We revisit this formulation and interpret the Wigner transform as an explicit realization of the DFS method, lifting functions from $\mathcal{S}\mathcal{O}(3)$ to ${\mathbb{T}}^3$. In this context, we analyze the Sobolev regularity loss induced by this lifting. Furthermore, we compare different Wigner transform implementations, examine additional symmetry enhancements, and observe that the direct method is often faster and more stable than the fast polynomial transform approaches.