# Fourier transform

(redirected from Fourier integral)
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Related to Fourier integral: Fourier series

## Fou·ri·er a·nal·y·sis

(fūr-ē-ā'),
a mathematical approximation of a function as the sum of periodic functions (sine and/or cosine waves) of different frequencies; a method of converting a function of time or space into a function of frequency; used in reconstruction of images in computed tomography and magnetic resonance imaging in radiology and in analysis of any kind of signal for its frequency content.

## Fourier transform (FT)

[fo̅o̅ryā′]
Etymology: Jean B.J. Fourier, French mathematician, 1768-1830; L, transformare, to change form
a mathematical procedure that separates out the frequency components of a signal from its amplitudes as a function of time, or vice versa.

## Fourier transform

A computational procedure used by MRI scanners to analyse and separate amplitude and phases of individual frequency components of the complex time varying signal, which allows spatial information to be reconstructed from the raw data.

## Fou·ri·er trans·form

(fūr-ē-ā' trans'fōrm)
A mathematical technique of dividing a time-varying function or signal into components at different frequencies, giving the phase and amplitude of each; used in computed tomography and magnetic resonance image reconstruction transformation.

## Fourier,

J.B.J., French mathematician and administrator, 1768-1830.
Fourier analysis - used in reconstruction of images in computed tomography and magnetic resonance imaging in radiology and in analysis of any kind of signal for its frequency content. Synonym(s): Fourier law; Fourier transform
Fourier law - Synonym(s): Fourier analysis
Fourier transform - Synonym(s): Fourier analysis
References in periodicals archive ?
Smith, A Hardy space for Fourier integral operators, J.
It was Dick Askey who realized that Wiener's treatment of the Fourier integrals [59] contains the key to q-extensions [8, 11, 41].
These solutions are based on the field representation in the form of Fourier integrals and series.
VIENER, Fourier Integral and Some Its Applications, (in Russian), Fizmatgiz, Moscow, 1963.
Based on five mini-courses and 15 of the lectures given at the workshop held in 2006 in Toronto by the Fields Institute, these papers address partial differential equations, geometric analysis, Fourier integral operators, localization operators, Gabor transforms, wavelet transforms, Rihaczek transforms and time-frequency analysis.
They uncover topological problems such as index theory and elliptic edge problems, then describe applications and related topics such as Fourier integral operators on singular manifolds, relative elliptic theory, the index of geometric operators on manifolds with cylindrical ends, the homotopy classification of elliptic operators and Lefschetz formulas.
The topics include local inverse scattering problems as a tool of perturbation analysis for resonance systems, Fourier integrals and a new representation of Maslov's canonical operator near caustics, the central limit theorem for linear eigenvalue statistics of the sum of independent random matrices of rank one, a homogenized model of oscillations of an elastic medium with small caverns filled with viscous incompressible fluid, and recovering a potential of the Sturm-Liouville problem from finite sets of spectral data.
He proceeds from the elementary theory of Fourier series and Fourier integrals to abstract harmonic analysis on locally compact abelian groups.
r[OMEGA]]([beta], [alpha]; x), the Fourier integrals in both the a and [beta] directions must be evaluated.
The Wiener algebra A(R) consists of those bounded continuous functions f which can represented as Fourier integrals of functions F 2 L1(R) :

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