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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?)
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    Why do you want to do this? The standard Fourier transform does not really apply to functions defined on a sphere. In this context, the analogues of Fourier modes are in fact precisely the spherical harmonics.2011-01-14
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    @Rahul Narain: I know what you mean, but Fourier space provides for example the big advantage that you can describe translations by mere phase factors while in spherical Harmonics you have to use rather messy expressions involving [Clebsch–Gordan coefficients](http://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients)...2011-01-14

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