[HN Gopher] Generalizations of Fourier Analysis
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Generalizations of Fourier Analysis
Author : mscharrer
Score : 65 points
Date : 2023-04-15 16:27 UTC (6 hours ago)
(HTM) web link (gabarro.org)
(TXT) w3m dump (gabarro.org)
| lixtra wrote:
| This sounds like a great direction for an advanced seminar in
| university.
| macrolocal wrote:
| Yosida is a great reference for this functional analysis.
|
| For a much broader generalization, albeit with expensive
| concepts, cf. Tannaka-Krein duality.
| mananaysiempre wrote:
| A couple of other things that AFAIK aren't special cases of the
| ones in the list:
|
| - The idempotent ("tropical") Fourier transform turns out to be
| the Legendre transform;
|
| - The fractional Fourier transform, known to physicists as the
| propagator of the quantum harmonic oscillator, is a pretty fun
| thing to consider;
|
| - The Fourier-Laplace transform on Abelian groups seems like a
| fairly straightforward extension of the idea of plugging in a
| complex frequency, but I haven't seen a textbook exposition (only
| an old article);
|
| - The non-linear Fourier transform (with Ki xi + Lij xi xj + ...,
| finite or infinite sum) seems impressively obscure (I know of a
| total of one book reference) but occurs in quantum field theory
| as the "n-loop" or "[?]-loop effective action";
|
| - The _odd_ (in the super sense) Fourier transform turns out to
| underpin stuff like the Hodge star on differential forms;
|
| - On a finite non-Abelian groups, the duality splits into two:
| every function on conjugacy classes is a linear combination of
| irreducible characters; every function on group is a linear
| combination of irreducible matrix elements; this is probably also
| doable on Lie groups but I'm too much of a wimp to learn the
| theory.
|
| (Also, generating functions should by all appearances be a fairly
| elementary chapter of the Fourier story, as electronic engineers
| with their "Z-transform" also realize, but I haven't seen that
| implemented convincingly in full.)
|
| See as well Baez's old issue of "This Week's Finds" where he
| started with sound and well all the way to spectra of Banach
| algebras and rings--as in Gelfand duality, algebraic geometry
| etc. (Can't seem to locate the specific issue now.) Of course
| there are also wavelets (there's even a Fields Medal for those
| now), but I don't know that they fit into the representation
| theory ideology (would be excited to be wrong!).
| enriquto wrote:
| > See as well Baez's old issue of "This Week's Finds"
|
| Well seen! TFA is certainly inspired by that very old Baez
| post, where he explained to Oz the many viewpoints of Fourier
| analysis. I was dismayed to see that all such viewpoints were
| algebraic in nature, requiring special structure in the base
| space, thus neglecting the fundamental case of a general
| manifold without symmetries. Now it seems that there are still
| missing generalizations!
| dustingetz wrote:
| what is an intuition for complex frequency?
| paulsutter wrote:
| Signals estimated by the FFT have two parameters: magnitude
| and phase. FFT results evade intuition because complex
| numbers are cartesian. If you convert them to polar
| coordinates they make more sense as magnitude and phase
|
| https://www.gaussianwaves.com/2015/11/interpreting-fft-
| resul...
|
| Note that complex numbers are merely convenient for working
| with two dimensional quantities. The square root of -1 is
| just math geek for orthogonal, and has nothing whatsoever to
| do with signals
| mananaysiempre wrote:
| (Note: GGP, not GP.) I meant complex frequency and not
| complex amplitude though.
| mananaysiempre wrote:
| As I meant it--of, not for.
|
| I referred to the idea that by plugging an imaginary
| frequency into the Fourier transform [ETA: the grown-up
| Fourier transform with the complex exponent, not the
| schoolboy cosine kludge], you get the Laplace transform, and
| while that changes the inverse Fourier transform in a
| different way, it's not hard to work out how specifically and
| obtain the inverse Laplace transform.
|
| Why you'd want to do that, I actually don't know how to
| explain convincingly. The post hoc rationalization is simple
| and more or less the reason people prefer the Laplace
| transform in signal processing: you still get a convolution
| theorem, but are now allowed to work with exponentially
| increasing functions, which standard Fourier theory (even the
| tempered distributions version) can't accomodate. But while
| that's useful from a toolbox standpoint, it isn't satisfying
| as motivation, I think.
|
| This is not the only way looking at the complex frequency
| plane turns out to be useful--there's a whole thing about
| doing complex analysis to response functions aka propagators
| --but there too I can't really say why you'd guess to look in
| that direction in the first place.
|
| What I mentioned was that this idea of Laplace as imaginary
| Fourier extends beyond the reals to the group setting at
| least to some extent, so it's not entirely an R-specific
| accident. Again, dunno why, I've explored this stuff a bit
| but am far from an expert.
| [deleted]
| hgomersall wrote:
| Not sure if you covered this, but I was super impressed by the
| general discrete Fourier transform:
| https://en.m.wikipedia.org/wiki/Discrete_Fourier_transform_o...
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