# Fourier Integral

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Related to Fourier Integral: Fourier series
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Fourier Integral

a formula for the decomposition of a nonperiodic function into harmonic components whose frequencies range over a continuous set of values.

If a function f(x) satisfies the Dirichlet condition on every finite interval and if the integral converges, then The formula was first introduced in 1811 by J. Fourier in connection with the solution of certain heat conduction problems but was proved later by other mathematicians.

Formula (1) can also be given in the form where In particular, for even functions where Formula (2) may be viewed as the limiting form of the Fourier series for functions with period 2T as T → ∞. Then, a(u) and b(u) are analogues of the Fourier coefficients of f(x).

Using complex numbers, we can replace formula (1) with Formula (1) can also be written as (simple Fourier integral).

If the integrals in formulas (2) and (3) diverge (seeIMPROPER INTEGRALS), then, in many cases, they nevertheless converge to f(x) if we use appropriate summability methods. The solution of many problems involves the use of Fourier integrals of functions of two and more variables.

### REFERENCE

Titchmarsh, E. Vvedenie v teoriiu integralov Fur’e. Moscow-Leningrad, 1948. (Translated from English.)
References in periodicals archive ?
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Tuan, Generalized convolutions for the Fourier integral transforms and applications, Journal of Siberian Federal Univ.
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Labate, Representation of Fourier integral operators using shearlets, J.
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