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!).