Fourier transform

A mathematical operation by which a function expressed in terms of one variable, x , may be related to a function of a different variable, s , in a manner that finds wide application in physics. The Fourier transform, F(s ), of the function f(x) is given by F(s) = f(x) exp(–2πixs) dx and f(x) = F(s) exp(2πixs) ds

The variables x and s are often called Fourier pairs. Many such pairs are useful, for example, time and frequency: the Fourier transform of an electrical oscillation in time gives the spectrum, or the power contained in it at different frequencies.

Fourier Transform

The Fourier transform of a function f(x) is the function f(x)

If f(x) is even, then g(u) takes the form

(the cosine transform), and if f(x) is odd, then g(u) takes the form

(the sine transform).

Formulas (1), (2), and (3) are invertible, that is, for even functions

for odd functions

and in general

To every operation on functions there corresponds an operation on their Fourier transforms that is frequently simpler than the operation on f(x). For example, the Fourier transform of f’(x) is iug(u). If

then g(u) = g₁(u)g₂(u). The Fourier transform of f(x + a) is e^iuag(u), and the Fourier transform of c₁f₁(x) is c₁g₁(u) + c₂g₂(u).

exists, then the integrals in formulas (1) and (6) converge in the mean (seeCONVERGENCE) and

(Plancherel’s theorem). Formula (8) is the extension of Parse-val’s formula to Fourier transforms (seePARSEVAL EQUALITY). In physical terms, formula (8) states that the energy of a certain vibration is the sum of the energies of its harmonic components. The map F:f(x) → g(u) is a unitary operator in the Hubert space of square integrable functions f(x), – ∞ < x < ∞. This operator can also be represented in the form

Under certain conditions on f(x) we have the Poisson formula

which is of use in the theory of theta functions.

If the function f(x) decreases sufficiently rapidly, then it is also possible to define its Fourier transform for certain complex values u = v + iw. For example, if

exists, then the Fourier transform is defined for |w| < a. There is a close connection between the complex Fourier transform and the two-sided Laplace transform

The Fourier transform operator can be extended to classes of functions larger than the class of integrable functions; for example, if (1 + |x|)^–1f(x) is integrable, then the Fourier transform of f(x) is given by formula (9). The operator can even be extended to certain classes of generalized functions (generalized functions with a slow rate of growth).

There exist generalizations of the Fourier transform. One of them makes use of various special funtions, such as Bessel functions. This generalization is fully developed in the representation theory of continuous groups. Another generalization is the Fourier-Stieltjes transform, which is frequently applied in, for example, probability theory; it is defined for every bounded nondecreasing function ϕ(x) by means of the Stieltjes integral

and is called the characteristic function of the distribution ϕ. According to the Bochner-Khinchin theorem, for g(u) to be representable by (10) it is necessary and sufficient that

for every choice of u₁, . . ., u_n, ξ₁, . . ., ξ_n.

First introduced in heat conduction theory, the Fourier transform has many applications in mathematics. For example, it is used in the solution of differential, difference, and integral equations and in the theory of special functions. Many branches of theoretical physics also make use of the transform. Thus, the Fourier transform provides the standard apparatus for quantum field theory and is widely applied in the method of Green’s functions for nonequilibrium problems of quantum mechanics and thermodynamics and in scattering theory.

REFERENCES

Sneddon, I. Preobrazovanie Fur’e. Moscow, 1955. (Translated from English.)
Vladimirov, V. S. Obobshchennye funktsii v matematicheskoi fizike. Moscow, 1976.

Fourier transform

[‚fu̇r·ē‚ā ′tranz‚fȯrm] (mathematics) For a function ƒ(t), the function F (x) equal to 1/√(2π) times the integral over t from -∞ to ∞ of ƒ(t) exp (itx).

Fourier transform

(mathematics)A technique for expressing a waveform as aweighted sum of sines and cosines.

Computers generally rely on the version known as discrete Fourier transform.

Named after J. B. Joseph Fourier (1768 -- 1830).

See also wavelet, discrete cosine transform.

Fourier transform

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. Synonym(s): Fourier transfer, Fourier transform

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 transformFourier law - Synonym(s): Fourier analysisFourier transform - Synonym(s): Fourier analysis

单词	fourier transform
释义	Fourier transform Fourier transform n. An operation that maps a function to its corresponding Fourier series or to an analogous continuous frequency distribution. [After Baron Jean Baptiste Joseph Fourier.] Fourier transform n (Mathematics) an integral transform, used in many branches of science, of the form F(x) = [1/√(2π)]∫e^ixyf(y)dy, where the limits of integration are from –∞ to +∞ and the function F is the transform of the function f Fourier transform Fourier transform A mathematical operation by which a function expressed in terms of one variable, x , may be related to a function of a different variable, s , in a manner that finds wide application in physics. The Fourier transform, F(s ), of the function f(x) is given by F(s) = f(x) exp(–2πixs) dx and f(x) = F(s) exp(2πixs) ds The variables x and s are often called Fourier pairs. Many such pairs are useful, for example, time and frequency: the Fourier transform of an electrical oscillation in time gives the spectrum, or the power contained in it at different frequencies. Fourier Transform The Fourier transform of a function f(x) is the function f(x) If f(x) is even, then g(u) takes the form (the cosine transform), and if f(x) is odd, then g(u) takes the form (the sine transform). Formulas (1), (2), and (3) are invertible, that is, for even functions for odd functions and in general To every operation on functions there corresponds an operation on their Fourier transforms that is frequently simpler than the operation on f(x). For example, the Fourier transform of f’(x) is iug(u). If then g(u) = g₁(u)g₂(u). The Fourier transform of f(x + a) is e^iuag(u), and the Fourier transform of c₁f₁(x) is c₁g₁(u) + c₂g₂(u). If exists, then the integrals in formulas (1) and (6) converge in the mean (seeCONVERGENCE) and (Plancherel’s theorem). Formula (8) is the extension of Parse-val’s formula to Fourier transforms (seePARSEVAL EQUALITY). In physical terms, formula (8) states that the energy of a certain vibration is the sum of the energies of its harmonic components. The map F:f(x) → g(u) is a unitary operator in the Hubert space of square integrable functions f(x), – ∞ < x < ∞. This operator can also be represented in the form Under certain conditions on f(x) we have the Poisson formula which is of use in the theory of theta functions. If the function f(x) decreases sufficiently rapidly, then it is also possible to define its Fourier transform for certain complex values u = v + iw. For example, if exists, then the Fourier transform is defined for \|w\| < a. There is a close connection between the complex Fourier transform and the two-sided Laplace transform The Fourier transform operator can be extended to classes of functions larger than the class of integrable functions; for example, if (1 + \|x\|)^–1f(x) is integrable, then the Fourier transform of f(x) is given by formula (9). The operator can even be extended to certain classes of generalized functions (generalized functions with a slow rate of growth). There exist generalizations of the Fourier transform. One of them makes use of various special funtions, such as Bessel functions. This generalization is fully developed in the representation theory of continuous groups. Another generalization is the Fourier-Stieltjes transform, which is frequently applied in, for example, probability theory; it is defined for every bounded nondecreasing function ϕ(x) by means of the Stieltjes integral and is called the characteristic function of the distribution ϕ. According to the Bochner-Khinchin theorem, for g(u) to be representable by (10) it is necessary and sufficient that for every choice of u₁, . . ., u_n, ξ₁, . . ., ξ_n. First introduced in heat conduction theory, the Fourier transform has many applications in mathematics. For example, it is used in the solution of differential, difference, and integral equations and in the theory of special functions. Many branches of theoretical physics also make use of the transform. Thus, the Fourier transform provides the standard apparatus for quantum field theory and is widely applied in the method of Green’s functions for nonequilibrium problems of quantum mechanics and thermodynamics and in scattering theory. REFERENCES Sneddon, I. Preobrazovanie Fur’e. Moscow, 1955. (Translated from English.) Vladimirov, V. S. Obobshchennye funktsii v matematicheskoi fizike. Moscow, 1976. Fourier transform [‚fu̇r·ē‚ā ′tranz‚fȯrm] (mathematics) For a function ƒ(t), the function F (x) equal to 1/√(2π) times the integral over t from -∞ to ∞ of ƒ(t) exp (itx). Fourier transform (mathematics)A technique for expressing a waveform as aweighted sum of sines and cosines. Computers generally rely on the version known as discrete Fourier transform. Named after J. B. Joseph Fourier (1768 -- 1830). See also wavelet, discrete cosine transform. Fourier transform 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. Synonym(s): Fourier transfer, Fourier transform 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 transformFourier law - Synonym(s): Fourier analysisFourier transform - Synonym(s): Fourier analysis AcronymsSeefaintThesaurusSeeFourier analysis
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