The fast fourier transform, which was invented by Tukey, significantly improves the speed of computation of discrete fourier transform.

the main application of fourier transform is the changing a function from frequency domain to time domain, laplaxe transform is the general form of fourier transform .

Fourier series and the Fourier transform

discrete fourier transformer uses digital signals whereas the fast fourier transform uses both analog and digital.

The Laplace transform is related to the Fourier transform, but whereas the Fourier transform expresses a function or signal as a series of modes ofvibration (frequencies), the Laplace transform resolves a function into its moments. Like the Fourier transform, the Laplace transform is used for solving differential and integral equations.