What is the difference (if any) between "fourier transform" and "SO(3) fourier transform"?


I searched on Google but couldn't find a satisfiable answer.


Thanks in advance :)



What is the difference (if any) between "fourier transform" and "SO(3) fourier transform"?
The usual (discrete) Fourier transform expands a function in the basis set $e^{i n\phi}$, $\phi\in(0,2\pi)$, $n\in\mathbb{Z}$. The $\text{SO}(3)$ Fourier transform uses as basis the [Wigner D-functions](https://en.wikipedia.org/wiki/Wigner_D-matrix) $D\_{\ell}^{m,n}$, which are an orthogonal basis for the rotation group,
$$f=\sum\_{\ell=1}^L\sum\_{m,n=-\ell}^\ell f\_{\ell,m,n}D\_\ell^{m,n}.$$
The integer $L$ is the degree of the transform. It is possible to rewrite this as a usual Fourier transform by expanding the Wigner-D function into a Fourier sum and then one can apply the Fast Fourier Transform algorithm, see [A Fast Fourier Algorithm on the Rotation Group](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.122.6676&rep=rep1&type=pdf). The complexity of this operation scales as $L^3\log L$.


This is an example of [noncommutative harmonic analysis](https://en.wikipedia.org/wiki/Noncommutative_harmonic_analysis).


