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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.26 The Fast Fourier Transform

If f : ImageImage is a periodic function with period 1, then one can obtain a great deal of useful information about f by calculating its Fourier coefficients (see THE FOURIER TRANSFORM [III.27] for a discussion Of why). This is true for both theoretical and practical reasons, and because of the latter it is highly desirable to have a good way of computing Fourier coefficients quickly. A method for doing this was discovered by Cooley and Tukey in 1965 (though it turned out that Gauss had anticipated them over 150 years ...

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