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Generalizations of Fourier analysis (2021)

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Re: Generalizations of Fourier analysis (2021)

#11

Post by enriquto »

mananaysiempre wrote: post #4

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 id…

> 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!

Re: Generalizations of Fourier analysis (2021)

#12

Post by antognini »

I am reminded somewhat of a line in Sanjeev Arora's lecture notes A Theorist's Toolkit:

"Sanjeev admits that he used to find Fourier transforms intimidating as a student. His fear vanished once he realized that it is a rewording of the following trivial idea:

If u_1, u_2, ..., u_n is an orthonormal basis of R^n then every vector v can be expressed as Sum_i alpha_i u_i where alpha_i = and Sum_i alpha_i^2 = |v|^2"

https://www.cs.princeton.edu/~arora/pubs/toolkit.pdf

Re: Generalizations of Fourier analysis (2021)

#14

Post by adgjlsfhk1 »

rthomas6 wrote: post #13

Ugh, I need to learn more math. How do I even start? I know multi variable calculus, I know the basics of linear algebra, and I know Fourier transforms. Yet this article is half gibberish to me.

imo abstract algebra is pretty much the gateway to most modern math. it well feel at first like it's a lot of machinery without purpose, but understanding groups rings and fields well opens the door to topology, advanced number theory and a bunch of the rest of math. the other option would be to learn some real and complex analysis, but imo the algebra side is where a lot more of the cool stuff is.

Re: Generalizations of Fourier analysis (2021)

#15

Post by jdhwosnhw »

mananaysiempre wrote: post #9
paulsutter wrote: post #6

Earlier quoted context omitted.

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.

From a physics perspective, complex frequency results in “evanescent waves” - ie, waves that decay rather than oscillate (technically a fully complex frequency of the form a+ib will both oscillate and decay)

Re: Generalizations of Fourier analysis (2021)

#16

Post by superb-owl »

> Notice that, even if their formulas look quite similar, the Fourier series is not a particular case of the Fourier transform. For example, a periodic function is never integrable over the real line unless it is identically zero. Thus, you cannot compute the Fourier transform of a periodic function.

Someone correct me if I'm wrong, but I do think the latter does generalize the former. I vaguely remember seeing it derived as essentially linking +/- infinity so the function is "periodic" on the real line. But I could be misremembering

This is the class I took, it's incredible: https://see.stanford.edu/Course/EE261/137

Re: Generalizations of Fourier analysis (2021)

#18

Post by ajkjk »

antognini wrote: post #12

I am reminded somewhat of a line in Sanjeev Arora's lecture notes A Theorist's Toolkit : "Sanjeev admits that he used to find Fourier transforms intimidating as a student. His fear vanished once he realized that it is a rewording of the following trivial idea: If u_1, u_2, ..., u_n is an orthonormal basis of R^n then every vector v can be expressed as Sum_i alpha_i u_i where alpha_i = and Sum_i alpha_i^2 = |v|^2" htt…

That's not a good enough statement to really summarize Fourier transforms, I think? It really just summarizes the idea of an orthogonal basis.

Re: Generalizations of Fourier analysis (2021)

#20

Post by nextaccountic »

mananaysiempre wrote: post #4

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 id…

> The non-linear Fourier transform (..) seems impressively obscure (I know of a total of one book reference)

Are those talking about the same thing?

[Information Transmission using the Nonlinear Fourier Transform, Part I: Mathematical Tools](https://arxiv.org/abs/1202.3653)

[Nonlinear Fourier Analysis](https://arxiv.org/abs/1201.5129)

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