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