For a function $f(r, \vartheta, \varphi)$ given in spherical coordinates, how can the Fourier transform be calculated best? Possible ideas:
- express $(r,\vartheta,\varphi)$ in cartesian coordinates, yielding a nonlinear argument of $f$
- express $\vec k,\vec r$ in the $e^{i\vec k\vec r}$ term in spherical coordinates, yielding a nonlinear exponent in $\vartheta$ and $\varphi$
- decompose $f$ into Spherical Harmonics and then change base to Fourier space, requiring the Fourier transform of the Spherical Harmonics (it is obviously not possible to calculate them using this very method..., can that be be found somewhere?)
Tobias, your notation makes it look like your function $f:\mathbb{R}^3\to\mathbb{R}$, in other words, it takes in points in three-dimensional space and spits out real numbers. In that case, as you note, it can be written as a function of $(x,y,z)$. So there doesn't seem to be any reason not to go with your first option. Maybe you could write down the function so I can see the difficulty. If on the other hand your function takes in points on the sphere $\{(x,y,z):\, x^2+y^2+z^2=1\}$, then it makes sense to use spherical harmonics. Your second option doesn't seem reasonable--if you want translation to correspond to phase shifts, then you need to integrate along lines in $\mathbb{R}^3$, and then after a change of coordinates you would be back in your first situation.