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Just a nit. The pair of equations the author showed at the beginning of the article are not the equations of the Fourier Transform and its inverse. The Transfor

Just a nit. The pair of equations the author showed at the beginning of the article are not the equations of the Fourier Transform and its inverse. The Transform is a continuous function operating on an infinite input. The equation for the Transform involves the use of the integral taken over +/- infinity. What is shown, using the summation operator, is a discrete form on the Transform where the input is a limited time series. A good alternative explanation can be found on Grant Sanderson's 3 Blue 1 Brown channel https://youtu.be/spUNpyF58BY?si=uqq2OOSATYcWmaG8 https://youtu.be/spUNpyF58BY?si=uqq2OOSATYcWmaG8

keithalewis 3y ago
Fourier transform can be defined for any locally compact abelian group. Integers modulo n is one such.
- penguin_booze 3y ago
  Given that monads are monoids in the category of endofunctors, I'm perfectly fine with it.
  - keithalewis 3y ago
    There is no Royal Road to mathematics. There is also no shortage of people convincing themselves they understand some particular aspect of it by watching a YouTube video. It is rare when a mathematician, or anyone, can write a book on a topic that no other expert in the field can top. Walter Rudin did that with Fourier analysis.
- ndriscoll 3y ago
  You actually don't need abelian. e.g. the group of 1d affine transformations T(a,b)(x) = ax+b gives a variation of a wavelet transform. But you no longer have 1-D irreducible representations, so your Fourier coefficients become operators instead of numbers or something like that.
  - keithalewis 3y ago
    What is Haar measure in this case?
    - ndriscoll 3y ago
      Apparently the left invariant measure is 1/a^2 dadb, and the right invariant measure is 1/a dadb. The book I have just sticks to left measures, and is an engineering book so it doesn't really get into details that a math book would (the whole group theory chapter is 31 pages out of a 1500 page book and the wavelet portion is ~2 pages, though it also has an 8 page section on wavelets in a previous chapter). It's very much a "what are groups, what are representations, and why do they matter for physics and signal processing" kind of thing. For reference it's Barrett & Myers Foundations of Image Science.

keithalewis 3y ago

Fourier transform can be defined for any locally compact abelian group. Integers modulo n is one such.

penguin_booze 3y ago
Given that monads are monoids in the category of endofunctors, I'm perfectly fine with it.
- keithalewis 3y ago
  There is no Royal Road to mathematics. There is also no shortage of people convincing themselves they understand some particular aspect of it by watching a YouTube video. It is rare when a mathematician, or anyone, can write a book on a topic that no other expert in the field can top. Walter Rudin did that with Fourier analysis.
ndriscoll 3y ago
You actually don't need abelian. e.g. the group of 1d affine transformations T(a,b)(x) = ax+b gives a variation of a wavelet transform. But you no longer have 1-D irreducible representations, so your Fourier coefficients become operators instead of numbers or something like that.
- keithalewis 3y ago
  What is Haar measure in this case?
  - ndriscoll 3y ago
    Apparently the left invariant measure is 1/a^2 dadb, and the right invariant measure is 1/a dadb. The book I have just sticks to left measures, and is an engineering book so it doesn't really get into details that a math book would (the whole group theory chapter is 31 pages out of a 1500 page book and the wavelet portion is ~2 pages, though it also has an 8 page section on wavelets in a previous chapter). It's very much a "what are groups, what are representations, and why do they matter for physics and signal processing" kind of thing. For reference it's Barrett & Myers Foundations of Image Science.