The Fourier transform, named for Jean Baptiste Joseph Fourier, is a frequency transform that decomposes a function into its sine and/or cosine parts (basis functions) for several frequencies. The result is a function in the frequency domain, representing the frequency spectrum of the original function.

The term Fourier transform is often taken to mean a linear operator taking a complex-valued function f (which may be, and in practice often is, real-valued) to a complex-valued function defined by

There are also discrete Fourier transforms and Fourier series.

The Fourier transform taking functions with domain A into functions with domain B may be:

• a Fourier series, in which A is an interval, such as [-π, &pi] and B is the set of all integers)

• a discrete Fourier transform in which A and B are segments of natural numbers - usually 0, ..., N − 1)

• a Continuous Fourier transform, in which A and B are the whole real line.

All the above are generalized by the Fourier transform on locally compact abelian topological groups, which is studied in harmonic analysis; here, A is the group and B is its dual group. This treatment also allows a general formulation of the convolution theorem, which relates Fourier transforms and convolutions.

The Fourier transform can be viewed as a special case of the Z-transform: the Fourier transform is the Z-transform evaluated at the unit circle in the complex space.

See the Fourier transform in action on the SETI at home project.

Actual implementations of Fourier transforms of arbitrary signals are computationally intensive. Such transforms are used in some types of RF modulation.

Kevin Cowtan's Book of Fourier offers an excellent introduction to the Fourier transform, especially regarding its application to X-ray crystallography.

See also Fourier inversion theorem.