The Fourier transform (English pronunciation: /ˈfɔərieɪ/), named after Joseph Fourier, is a mathematical transformation employed to transform signals between time (or spatial) domain and frequency domain, which has many applications in physics and engineering. It is reversible, being able to transform from either domain to the other. The term itself refers to both the transform operation and to the function it produces.
In the case of a periodic function over time (for example, a continuous but not necessarily sinusoidal musical sound), the Fourier transform can be simplified to the calculation of a discrete set of complex amplitudes, called Fourier series coefficients. They represent the frequency spectrum of the original time-domain signal. Also, when a time-domain function is sampled to facilitate storage or computer-processing, it is still possible to recreate a version of the original Fourier transform according to the Poisson summation formula, also known as discrete-time Fourier transform. See alsoFourier analysis and List of Fourier-related transforms.
There are several common conventions for defining the Fourier transform 
of an integrable function 
(Kaiser 1994, p. 29), (Rahman 2011, p. 11). This article will use the following definition:
of an integrable function (Kaiser 1994, p. 29), (Rahman 2011, p. 11). This article will use the following definition:, for any real number ξ.
When the independent variable x represents time (with SI unit of seconds), the transform variable ξ represents frequency (in hertz). Under suitable conditions, 
is determined by 
via the inverse transform:
is determined by via the inverse transform:, for any real number x.
The statement that can be reconstructed from is known as the Fourier inversion theorem, and was first introduced in Fourier's Analytical Theory of Heat (Fourier 1822, p. 525), (Fourier & Freeman 1878, p. 408), although what would be considered a proof by modern standards was not given until much later (Titchmarsh 1948, p. 1). The functions and often are referred to as a Fourier integral pair or Fourier transform pair (Rahman 2011, p. 10).
can be reconstructed from is known as the Fourier inversion theorem, and was first introduced in Fourier's Analytical Theory of Heat (Fourier 1822, p. 525), (Fourier & Freeman 1878, p. 408), although what would be considered a proof by modern standards was not given until much later (Titchmarsh 1948, p. 1). The functions and often are referred to as a Fourier integral pair or Fourier transform pair (Rahman 2011, p. 10).
For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ, see Other conventions and Other notations below. The Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum.
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