Let $f$ and $g$ be two functions defined over the 2d sphere $\mathbb{S}^2$.
The convolution between $f$ and $g$ is defined as a function $f * g$ over the space $SO(3)$ of 3d rotations as $$(f*g)(R) = \int_{\mathbb{S}^2} f(R^{-1}x) g(x) \mu(dx)$$ where $\mu$ is the Haar measure on $\mathbb{S}^2$.
One can expand $f$ and $g$ in the orthonormal basis of spherical harmonics $Y_n^m$ for $n\geq0$ and $|m|\leq n$ such that $f = \sum \hat{f}_n^m Y_n^m$ where $\hat{f}_n^m = \langle f, Y_n^m \rangle$ (similarly for $g$).
Is there a Fourier relation for the Fourier transform of $f*g$ as is the case for instance for functions over $\mathbb{R}$, for which the Fourier transform of the convolution is the product of the Fourier transforms?
Note that this supposes to have a notion of Fourier transform for functions on $SO(3)$, which is provided by the Wigner D-matrix instead of the spherical harmonics as the Fourier basis.
