Is it possible to make an algorithm faster than the fast Fourier transform to calculate the discrete Fourier transform (is there proofs for or against it)?
OR, a one that only approximates the discrete Fourier transform, but is faster than $O(NlogN)$ and still gives about reasonable results?
Additional requirements:
1) Let's leave the quantum computing out
2) I don't mean faster in sense of how its implemented for some specific hardware, but in the "Big-O notation sense", that it would ran e.g. in linear time.
Sorry for my english
There is an algorithm called sparse fast Fourier transform (sFFT), which is faster than FFT algorithms when the Fourier coefficients are sparse.