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Published online by Cambridge University Press: 16 May 2025
Fourier developed his groundbreaking expansion methods at first as a tool for analyzing heat flow. From this application (and originally with little mathematical rigor) these expansions evolved rapidly to become one of the foremost tools in present day applied mathematics. The three main versions we describe in more detail are Fourier Transform (FT), Fourier Series (FS) and Discrete Fourier Transform (DFT), and how these are related to each other. Each case amounts to a transform pair – allowing one to move either way between physical and transform variables. The typical purpose of applying transforms is that certain operations are simpler in one of the spaces than in the other. This overview is followed by a discussion of the Fast Fourier Transform (FFT) algorithm, which is a computationally rapid way to carry out the DFT. This algorithm (by Cooley and Tukey, in 1965) caused one of the greatest computational advances of all time. The applications of this algorithm are far reaching.
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