[openasocket]

Yeah, I've always found generalized Fourier transforms and harmonic analysis to be really interesting! For those interested in reading more, some of the foundational work is the notion of a Haar measure and the Pontryagin duality. Basically, for any locally compact abelian group you can define a Fourier transform, and this Fourier transform is unique up to multiplicative identity. If you use this definition on the group of real numbers under addition you get the continuous Fourier transform, and if you use it on the group of integers under addition you get the discrete Fourier transform! And you can define the Fourier transform for more exotic groups, like on elliptic curves. In fact, this is how Shor's algorithm is able to factor large numbers, solve the discrete log problem, and break elliptic curve cryptography all in one algorithm: the algorithm is essentially just taking the Fourier transform of said function and measuring. Thanks to harmonic analysis, you can define a Fourier transform for elliptic curves and just plug that into Shor's algorithm

[stainforth]

Will harmonic analysis play into P!=NP then, if its usable in quantum math?

[openasocket]

I mean it could be, but not in the way you're talking about. There's a common misconception that quantum algorithms being able to factor integers and solve the discrete log problem in polynomial time has some implications for P vs NP. However, integer factoring and the discrete log problems are NOT NP complete problems, and there is no quantum algorithm published that can solve an NP complete problem in polynomial time. It's actually an open problem if quantum computers, in terms of computability classes, are any more powerful than classical computers (I think even assuming P!=NP, but I'm not a computing theory person haven't looked into more recent work on it).