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30 lectures on Fourier transforms and not one mention of the Fourier transform over compact groups? (Which is arguably the most generalized/abstract formulatio

30 lectures on Fourier transforms and not one mention of the Fourier transform over compact groups? (Which is arguably the most generalized/abstract formulation of the concept of a Fourier transform.) A lot of recent research has been done on FFTs over groups like the rotation group and the symmetric group, and this work has lead to significant progress on problems that were previously considered intractable.

jsprogrammer 10y ago
Do you know of a link that does host such mentions?
- Xcelerate 10y ago
  I've been using the book Engineering Applications of Noncommutative Harmonic Analysis by by Chirikjian and Kyatkin (https://www.amazon.com/dp/0849307481/ref=wl_it_dp_o_pC_nS_ttl https://www.amazon.com/dp/0849307481/ref=wl_it_dp_o_pC_nS_tt...). It's very well-written, and I believe an updated version of the book was recently released.
JadeNB 10y ago
> the Fourier transform over compact groups … is arguably the most generalized/abstract formulation of the concept of a Fourier transform. A good-spirited correction: anyone who knows enough to know this much generality knows that calling anything mathematical the "most generalised" version of a concept is begging to be corrected. Source: I do harmonic analysis on non-compact groups, and suspect that the folks over at the n-category café probably regard my kind of work as dangerously pedestrian (see, for example, https://golem.ph.utexas.edu/category/2010/11/integral_transforms_and_pullpu.html https://golem.ph.utexas.edu/category/2010/11/integral_transf...).
- Xcelerate 10y ago
  Haha, fair point (although in my defense, I did qualify my statement with "arguably"). That's interesting that you do harmonic analysis on non-compact groups; I've been meaning to learn more about that actually. I've been working on group synchronization problems related to the ideas in this paper (http://arxiv.org/pdf/1505.03840.pdf http://arxiv.org/pdf/1505.03840.pdf), which briefly mentions how one might extend the concept to non-compact groups, but doesn't elaborate further.
  - JadeNB 10y ago
    Sorry for the delayed response; I'm "submitting too fast". > I did qualify my statement with "arguably" I always tell my students—and, though it's a joke, I think that it's more true than casual consideration might suggest—that the correct answer (though not one that they're allowed to use on tests!) to any question in mathematics is "it depends." The more elementary the question, the more profound it sounds that you can see angles from which different answers can be correct. (Of course, backing this up with actual profound re-interpretations is the hard part. :-) ) > That's interesting that you do harmonic analysis on non-compact groups; I've been meaning to learn more about that actually. I've been working on group synchronization problems related to the ideas in this paper (http://arxiv.org/pdf/1505.03840.pdf http://arxiv.org/pdf/1505.03840.pdf), which briefly mentions how one might extend the concept to non-compact groups, but doesn't elaborate further. As a decidedly pure mathematician, that's outside my wheelhouse, although I did have the pleasure of having some of the applications explained to me by one of the authors of one of the papers on cryo-electron microscopy cited in the linked article. I'm afraid that the idea of embedding a non-compact group in a larger, compact group is foreign to my way of thinking about things, so (in case you were implicitly asking) unfortunately I can't shed any light there.

jsprogrammer 10y ago

Do you know of a link that does host such mentions?

Xcelerate 10y ago
I've been using the book Engineering Applications of Noncommutative Harmonic Analysis by by Chirikjian and Kyatkin (https://www.amazon.com/dp/0849307481/ref=wl_it_dp_o_pC_nS_ttl https://www.amazon.com/dp/0849307481/ref=wl_it_dp_o_pC_nS_tt...). It's very well-written, and I believe an updated version of the book was recently released.

JadeNB 10y ago

> the Fourier transform over compact groups … is arguably the most generalized/abstract formulation of the concept of a Fourier transform. A good-spirited correction: anyone who knows enough to know this much generality knows that calling anything mathematical the "most generalised" version of a concept is begging to be corrected. Source: I do harmonic analysis on non-compact groups, and suspect that the folks over at the n-category café probably regard my kind of work as dangerously pedestrian (see, for example, https://golem.ph.utexas.edu/category/2010/11/integral_transforms_and_pullpu.html https://golem.ph.utexas.edu/category/2010/11/integral_transf...).

Xcelerate 10y ago
Haha, fair point (although in my defense, I did qualify my statement with "arguably"). That's interesting that you do harmonic analysis on non-compact groups; I've been meaning to learn more about that actually. I've been working on group synchronization problems related to the ideas in this paper (http://arxiv.org/pdf/1505.03840.pdf http://arxiv.org/pdf/1505.03840.pdf), which briefly mentions how one might extend the concept to non-compact groups, but doesn't elaborate further.
- JadeNB 10y ago
  Sorry for the delayed response; I'm "submitting too fast". > I did qualify my statement with "arguably" I always tell my students—and, though it's a joke, I think that it's more true than casual consideration might suggest—that the correct answer (though not one that they're allowed to use on tests!) to any question in mathematics is "it depends." The more elementary the question, the more profound it sounds that you can see angles from which different answers can be correct. (Of course, backing this up with actual profound re-interpretations is the hard part. :-) ) > That's interesting that you do harmonic analysis on non-compact groups; I've been meaning to learn more about that actually. I've been working on group synchronization problems related to the ideas in this paper (http://arxiv.org/pdf/1505.03840.pdf http://arxiv.org/pdf/1505.03840.pdf), which briefly mentions how one might extend the concept to non-compact groups, but doesn't elaborate further. As a decidedly pure mathematician, that's outside my wheelhouse, although I did have the pleasure of having some of the applications explained to me by one of the authors of one of the papers on cryo-electron microscopy cited in the linked article. I'm afraid that the idea of embedding a non-compact group in a larger, compact group is foreign to my way of thinking about things, so (in case you were implicitly asking) unfortunately I can't shed any light there.