Generalizations of Fourier analysis (2021)
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Generalizations of Fourier analysis (2021)
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Re: Generalizations of Fourier analysis (2021)
#2This sounds like a great direction for an advanced seminar in university.
Re: Generalizations of Fourier analysis (2021)
#3Yosida is a great reference for this functional analysis.
For a much broader generalization, albeit with expensive concepts, cf. Tannaka-Krein duality.
Re: Generalizations of Fourier analysis (2021)
#4A 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!).
Re: Generalizations of Fourier analysis (2021)
#5A 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 id…
what is an intuition for complex frequency?
Re: Generalizations of Fourier analysis (2021)
#6A 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 id…
what is an intuition for complex frequency?
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
Re: Generalizations of Fourier analysis (2021)
#7A 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 id…
what is an intuition for complex frequency?
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.
Re: Generalizations of Fourier analysis (2021)
#8A 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 id…
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...
Re: Generalizations of Fourier analysis (2021)
#9Earlier quoted context omitted.
what is an intuition for complex frequency?
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…
(Note: GGP, not GP.) I meant complex frequency and not complex amplitude though.
Re: Generalizations of Fourier analysis (2021)
#10A 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 id…
what is an intuition for complex frequency?
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