The DFT (discrete Fourier transform) is basically a digital sampling (ie approximation) of the analog Fourier transform, and as such it remains finite throughout the domain.
http://en.wikipedia.org/wiki/Discrete_Fourier_transform
A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform and its inverse, for eg, a particular problem.
http://en.wikipedia.org/wiki/Fast_Fourier_transform
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