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Generalized Fourier Integral


The so-called generalized Fourier integral is a pair of integrals--a "lower Fourier integral" and an "upper Fourier integral"--which allow certain complex-valued functions f to be decomposed as the sum of integral-defined functions, each of which resembles the usual Fourier integral associated to f and maintains several key properties thereof.

Let x be a real variable, let alpha=sigma+itau be a complex variable, and let f=f(x) be a function for which |f(x)|<=A·exp(tau_-x) as x->infty, for which |f(x)|<=B·exp(tau_+x) as x->-infty, and for which f(x)exp(-tau_0x) has an analytic Fourier integral where here, tau_-<tau_0<tau_x are finite real constants. Next, define the upper and lower generalized Fourier integrals F_+(alpha) and F_-(alpha) associated to f, respectively, by

 F_+(alpha)=1/(sqrt(2pi))int_0^inftyf(x)e^(ialphax)dx
(1)

and

 F_-(alpha)=1/(sqrt(2pi))int_(-infty)^0f(x)e^(ialphax)dx
(2)

on the complex regions tau>tau_- and tau<tau_+, respectively. Then, for a>tau_- and b<tau_+,

 f(x)=1/(sqrt(2pi))int_(-infty+ia)^(infty+ia)F_+(alpha)e^(-ialphax)dalpha+1/(sqrt(2pi))int_(-infty+ib)^(infty+ib)F_-(alpha)e^(-ialphax)dalpha
(3)

where the first integral summand equals f(x) for x>0 and is zero for x<0 while the second integral summand is zero for x>0 and equals f(x) for x<0. The decomposition () is called the generalized Fourier integral corresponding to f.

Note that some literature defines the upper and lower integrals F_+ and F_- with multiplicative constants different from (2pi)^(-1/2), whereby the identity in () may look slightly different.


See also

Fourier Transform, Generalized Fourier Series, Integral Transform, Laplace Transform

This entry contributed by Christopher Stover

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References

Linton, C. M. and McIver, P. Handbook of Mathematical Techniques for Wave/Structure Interactions. Boca Raton, FL: CRC Press, 2001.Noble, B. Methods Based on the Wiener-Hopf Technique For the Solution of Partial Differential Equations. Belfast, Northern Ireland: Pergamon Press, 1958.

Cite this as:

Stover, Christopher. "Generalized Fourier Integral." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/GeneralizedFourierIntegral.html

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