Timeline for Fourier transform of functions mapping manifolds, is there a definition?
Current License: CC BY-SA 4.0
13 events
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| Dec 24, 2022 at 13:24 | comment | added | user49822 | Another option for the case where $f$ is differentiable is to form the derivative of $f$ as a map $f':\mathbb{R}\to so_3^n$ to the lie algebra of $SO(3)^n$, and as lie algebras are vector spaces you can try to compute the Fourier transform of $f'$ | |
| Dec 5, 2022 at 20:46 | review | Close votes | |||
| Dec 20, 2022 at 3:07 | |||||
| Dec 5, 2022 at 16:07 | comment | added | user8469759 | @Kostya_I I thought of that but I wonder if there's something like Banach algebra style. With that I mean you can derive the fourier transform in $L^1$ by finding the set of all homomorphism $\left\{ \phi \right\}$. Namely you go with $\phi(xy) = \phi(x)\phi(y)$ a bit of manipulations and you end up with the fourier transform. Don't know if this make sense but this is what I was going for. Homomrphism for groups are well defined, I thought maybe there's something there? | |
| Dec 5, 2022 at 16:00 | comment | added | Kostya_I | Well, $SO(3)$ is not a vector space, but it embeds into $\mathbb{R}^9$ as $3\times 3$ matrices. You can just think of you functions as taking values in $\mathbb{R}^{9n}$, ignoring that the actual range is smaller. It is not clear without further context why this is this inadequate. | |
| Dec 5, 2022 at 15:31 | comment | added | LSpice | 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. | |
| Dec 5, 2022 at 15:06 | comment | added | user8469759 | Cause I have a time series data of rotations expressed as matrices/quaternions or equivalent. Because of that target space I abstracted the question in the one I've asked. Just wondering if someone has thought about it really. | |
| Dec 5, 2022 at 15:04 | comment | added | Kostya_I | Could you spell out why you would apply a Fourier transform, i.e., what properties you want it to satisfy? | |
| Dec 5, 2022 at 14:28 | history | edited | YCor | CC BY-SA 4.0 |
formatting, added tags
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| Dec 5, 2022 at 13:50 | comment | added | user8469759 | Hopefully is clearer now. | |
| Dec 5, 2022 at 13:50 | history | edited | user8469759 | CC BY-SA 4.0 |
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| Dec 5, 2022 at 11:45 | comment | added | user8469759 | I guess but because lie groups have a bit of more structure I thought it was more likely to get an answer. | |
| Dec 5, 2022 at 11:39 | comment | added | Overflowian | Maybe functions "targeting" manifolds would be a better choice of words. | |
| Dec 5, 2022 at 11:22 | history | asked | user8469759 | CC BY-SA 4.0 |