Fourier transform

Etymology
Named after French mathematician and physicist, who initiated the study of what is now harmonic analysis.

Noun

 * 1)  A particular integral transform that when applied to a function of time (such as a signal), converts the function to one that plots the original function's frequency composition; the resultant function of such a conversion.
 * 2) * 2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, 1859, page 1,
 * The trigonometric sums of $$\mathcal{G}^F$$ are thus (up to a scalar) the Fourier transforms of the characteristic functions of the $$G^F\!\!$$-orbits of $$\mathcal{G}^F$$.
 * 1) * 2012, David Brandwood, Fourier Transforms in Radar and Signal Processing,, 2nd Edition, page 1,
 * The Fourier transform is a valuable theoretical technique, used widely in fields such as applied mathematics, statistics, physics, and engineering.
 * The trigonometric sums of $$\mathcal{G}^F$$ are thus (up to a scalar) the Fourier transforms of the characteristic functions of the $$G^F\!\!$$-orbits of $$\mathcal{G}^F$$.
 * 1) * 2012, David Brandwood, Fourier Transforms in Radar and Signal Processing,, 2nd Edition, page 1,
 * The Fourier transform is a valuable theoretical technique, used widely in fields such as applied mathematics, statistics, physics, and engineering.

Usage notes

 * Like the term itself, Fourier transform can mean either the integral operator that converts a function, or the function that is the end product of the conversion process.
 * The Fourier transform of a function $$f$$ is traditionally denoted $$\hat f$$. Several other notations are also used.
 * There are also several different conventions used when it comes to defining the Fourier transform and its inverse for an integrable function $$f: \mathbb{R} \to \mathbb{C}$$. (The two are often defined together to highlight their connectedness.)
 * One form of this definition pair is:
 * $$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{-2\pi i x \xi}\,dx$$
 * $$f(x) = \int_{-\infty}^{\infty} \hat f(\xi)\ e^{2 \pi i x \xi}\,d\xi$$,
 * where the exponent (including its sign) reflects a convention in electrical engineering to use $$f(x) = e^{2 \pi i \xi_0 x}$$ for a signal with initial phase 0 and frequency $$\xi_0$$.

Derived terms

 * (CFT)
 * (DFT)
 * (FFT)
 * (FTIR)
 * (FTIR)

Translations

 * Dutch: Fouriertransformatie, Fourier-transformatie
 * Finnish: Fourier-muunnos
 * French:
 * Galician: transformada de Fourier
 * German: Fouriertransformation, Fourier-Transformation
 * Greek: μετασχηματισμός Φουριέ
 * Italian: trasformata di Fourier
 * Japanese: フーリエ変換
 * Persian: تبدیل فوریه
 * Polish: transformacja Fouriera
 * Romanian: tranformare Fourier
 * Russian: преобразова́ние Фурье́
 * Spanish: transformada de Fourier
 * Swedish: Fouriertransform