A couple of other things that AFAIK aren’t special cases of the ones in the list:<p>- The idempotent (“tropical”) Fourier transform turns out to be the Legendre transform;<p>- The fractional Fourier transform, known to physicists as the propagator of the quantum harmonic oscillator, is a pretty fun thing to consider;<p>- 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);<p>- 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”;<p>- The <i>odd</i> (in the super sense) Fourier transform turns out to underpin stuff like the Hodge star on differential forms;<p>- 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.<p>(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.)<p>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!).
I am reminded somewhat of a line in Sanjeev Arora's lecture notes <i>A Theorist's Toolkit</i>:<p>"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:<p>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 = <v, u_i> and Sum_i alpha_i^2 = |v|^2"<p><a href="https://www.cs.princeton.edu/~arora/pubs/toolkit.pdf" rel="nofollow">https://www.cs.princeton.edu/~arora/pubs/toolkit.pdf</a>
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.
> 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.<p>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<p>This is the class I took, it's incredible: <a href="https://see.stanford.edu/Course/EE261/137" rel="nofollow">https://see.stanford.edu/Course/EE261/137</a>
Let's not forget the graph Fourier transform: <a href="https://en.wikipedia.org/wiki/Graph_Fourier_transform" rel="nofollow">https://en.wikipedia.org/wiki/Graph_Fourier_transform</a>
I asked ChatGPT to explain the intro section of this article like I'm five:<p>"""
Fourier analysis is like playing with music notes. We have different ways of looking at these notes, like how high or low they are, or how long they last. In the same way, we have different ways of looking at signals and how they change over time or space.<p>Just like how we can make different types of music by combining different notes in different ways, we can make different types of signals by combining different types of Fourier analysis.<p>There are four classic ways to do Fourier analysis, and each one is like a different way of looking at the signal. They're like different types of glasses we can put on to see the signal more clearly.<p>But there are also other ways to do Fourier analysis, like looking at signals on different types of spaces or using different types of math. It's like having even more types of glasses to look at the signal!<p>So when you ask a mathematician what Fourier analysis is, they might give you a different answer depending on which type of glasses they like to wear.
"""
Yosida is a great reference for this functional analysis.<p>For a much broader generalization, albeit with expensive concepts, cf. Tannaka-Krein duality.