Fourier transform


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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 eiuag(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, . . ., un, ξ1, . . ., ξn.

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 a weighted 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.
References in periodicals archive ?
The seasonality and diurnal variation of wind speed are observed by applying Fast Fourier Transform on NOAA wind data in New York and Austin.
Sangwine, "Connecting spatial and frequency domains for the quaternion Fourier transform," Applied Mathematics and Computation, vol.
de Rooij, "Miniaturized time scanning Fourier transform spectrometer based on silicon technology," Optics Letters, vol.
(2) Taking the Fourier transform of the signal obtained in step 1, (6) is used with a suitable predefined value for [gamma]; then we get [H.sub.p](f) = H(f).
Noticing that the phase [PHI]([phi]) is a periodic function of [phi], we apply Fourier transform to (1) and obtain
It should be noticed that when the parameters of the SAFT are chosen to be the special cases of the SAFT, the derived results reduce to the classical results of Fourier transform domain, fractional Fourier transform domain, and linear canonical transform domains.
We also study the relationship among the QWVD-LCT, the quaternion windowed Fourier transform, and the continuous quaternion Fourier transform.
Figure 5.1(b),(c) illustrates the error of Algorithm 2 using the butterfly fast Fourier transform (BSFFT) and the nonequispaced fast Fourier transform (NFFT), respectively.
Analysis on measuring accuracy of Fourier transform profilometry due to different filtering window, Laser Journal, 24(4), 43-45.
We observe that the function p [fleche diestra] [s.sub.p] from (2/3, 2) into (0,1) is bijective; with this in mind one can combine Szasz's results with our main theorem in order to give p-summability of the periodic Fourier transform on the interval (2/3, ([infinito]).
Bracewell, The Fourier Transform & Its Applications, McGgraw-Hill, New York, NY, USA, 1965.

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