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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

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Quantum Fourier Transform: $O(\log^2 N)$ – draks ... Nov 20 '12 at 20:01
Fastest currently implemented algorithm is fftw: fastest Fourier transform in the west. It's $O(n\log n)$ though. – icurays1 Nov 20 '12 at 20:07
Mmm, ok, additional requirement: one that could be ran on computers we have today. :) – just Nov 20 '12 at 20:07
This seems in the right direction. – WimC Nov 20 '12 at 20:16
You've seen the Wikipedia page? "All known FFT algorithms require $\Theta(N \log N)$ operations... although there is no known proof that a lower complexity score is impossible." It would be interesting if we find out that Wikipedia is out of date, of course. – Rahul Nov 20 '12 at 20:24
up vote 1 down vote accepted

There is an algorithm called sparse fast Fourier transform (sFFT), which is faster than FFT algorithms when the Fourier coefficients are sparse.

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This appears to be partly the same that was pointed out by WimC (the paper he mentions could be found from the page you provided). I will accept your answer. Thanks also for WimC! – just Nov 20 '12 at 22:05

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