Editing Fourier analysis (section)

===(Continuous) Fourier transform===
{{main|Fourier transform}}

Most often, the unqualified term '''Fourier transform''' refers to the transform of functions of a continuous [[real number|real]] argument, and it produces a continuous function of frequency, known as a ''frequency distribution''. One function is transformed into another, and the operation is reversible. When the domain of the input (initial) function is time (<math>t</math>), and the domain of the output (final) function is [[frequency|ordinary frequency]], the transform of function <math>s(t)</math> at frequency <math>f</math> is given by the [[complex number]]''':'''

:<math>S(f) = \int_{-\infty}^{\infty} s(t) \cdot e^{- i2\pi f t} \, dt.</math>

Evaluating this quantity for all values of <math>f</math> produces the ''frequency-domain'' function. Then <math>s(t)</math> can be represented as a recombination of [[complex exponentials]] of all possible frequencies''':'''

:<math>s(t) = \int_{-\infty}^{\infty} S(f) \cdot e^{i2\pi f t} \, df,</math>

which is the inverse transform formula. The complex number, <math>S(f),</math> conveys both amplitude and phase of frequency <math>f.</math>

See [[Fourier transform]] for much more information, including''':'''
* conventions for amplitude normalization and frequency scaling/units
* transform properties
* tabulated transforms of specific functions
* an extension/generalization for functions of multiple dimensions, such as images.