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Fourier transform

The Fourier transform decomposes a function of time (a ''signal'') into the frequencies that make it up, similarly to how a musical chord can be expressed as the amplitude (or loudness) of its constituent notes. The Fourier transform of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is called the ''frequency domain representation'' of the original signal. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, the domain of the original function is commonly referred to as the ''time domain''. For many functions of practical interest one can define an operation that reverses this: the ''inverse Fourier transformation'', also called ''Fourier synthesis'', of a frequency domain representation combines the contributions of all the different frequencies to recover the original function of time.
Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency,〔Up to an imaginary constant factor whose magnitude depends on what Fourier transform convention is used.〕 so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain. Concretely, this means that any linear time-invariant system, such as a filter applied to a signal, can be expressed relatively simply as an operation on frequencies.〔The Laplace transform is a generalization of the Fourier transform that offers greater flexibility for many such applications.〕 After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to almost all areas of modern mathematics.
Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.
The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory.〔Depending on the application a Lebesgue integral, distributional, or other approach may be most appropriate.〕 For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint.〔 provides solid justification for these formal procedures without going too deeply into functional analysis or the theory of distributions.〕 The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of space to a function of momentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either space or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued.〔In relativistic quantum mechanics one encounters vector-valued Fourier transforms of multi-component wave functions. In quantum field theory operator-valued Fourier transforms of operator-valued functions of spacetime are in frequent use, see for example .〕 Still further generalization is possible to functions on groups, which, besides the original Fourier transform on or (viewed as groups under addition), notably includes the discrete-time Fourier transform (DTFT, group = ), the discrete Fourier transform (DFT, group = ) and the Fourier series or circular Fourier transform (group = , the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The fast Fourier transform (FFT) is an algorithm for computing the DFT.
==Definition==
There are several common conventions for defining the Fourier transform $\hat$ of an integrable function $f : \mathbb R \rightarrow \mathbb C$ , . This article will use the following definition:
:$\hat\left(\xi\right) = \int_^\infty f\left(x\right)\ e^\,dx,$   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, $f$ is determined by $\hat f$ via the inverse transform:
:$f\left(x\right) = \int_^\infty \hat f\left(\xi\right)\ e^\,d\xi,$   for any real number ''x''.
The statement that $f$ can be reconstructed from $\hat f$ is known as the Fourier inversion theorem, and was first introduced in Fourier's ''Analytical Theory of Heat'' , , although what would be considered a proof by modern standards was not given until much later . The functions $f$ and $\hat$ often are referred to as a ''Fourier integral pair'' or ''Fourier transform pair'' .
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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