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$\DeclareMathOperator\SO{SO}$I have a problem which boils down to the analysis of functions of the form $$ f : \mathbb{R} \to \SO(3)^n $$ Since $\SO(3)$ is a compact group so is $\SO(3)^n$. Now if instead of having such a mapping I'd had something from $\mathbb{R} \to \mathbb{R}^n$ I would've applied the Fourier transform componentwise (which I think is a valid definition of Fourier transform for such functions, please correct me if I am wrong).

However I cannot do the same thing in the case I am given because $\SO(3)$ is not even a vector space in the first place. I wonder if there's any definition of Fourier Transform that can be applied to my case.

Is there a definition I could use? or something someone came up with that might be useful in this case?

I know for functions $f: \mathcal{M} \to \mathbb{R}$ where $\mathcal{M}$ is a Riemannian manifold there's a definition of Fourier transform, relying on the Laplace-Beltrami operator. Can this be generalized if the target space is another manifold?

Thank you

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    $\begingroup$ Maybe functions "targeting" manifolds would be a better choice of words. $\endgroup$
    – Overflowian
    Commented Dec 5, 2022 at 11:39
  • $\begingroup$ I guess but because lie groups have a bit of more structure I thought it was more likely to get an answer. $\endgroup$
    – user8469759
    Commented Dec 5, 2022 at 11:45
  • $\begingroup$ Hopefully is clearer now. $\endgroup$
    – user8469759
    Commented Dec 5, 2022 at 13:50
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    $\begingroup$ Could you spell out why you would apply a Fourier transform, i.e., what properties you want it to satisfy? $\endgroup$
    – Kostya_I
    Commented Dec 5, 2022 at 15:04
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    $\begingroup$ The problem is that askng for a Fourier transform of an $\operatorname{SO}(3)^n$-valued function is quite similar to asking "I have some data; should I use the Fourier transform, the Laplace transform, or the Riesz transform?" Without context, there's no way to give an appropriate answer. Usually the Fourier transform is meant to exploit symmetries in the domain, so it's less clear to me what to do with symmetries on the target. $\endgroup$
    – LSpice
    Commented Dec 5, 2022 at 15:31

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