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Plancherel's Theorem

Plancherel's theorem states that the integral of the squared modulus of a function is equal to the integral of the squared modulus of its spectrum. It corresponds to Parseval's theorem for Fourier series. It is sometimes also known as Rayleigh's theory, since it was first used by Rayleigh (1889) in the investigation of blackbody radiation. In 1910, Plancherel first established conditions under which the theorem holds (Titchmarsh 1924; Bracewell 1965, p. 113).

In other words, let E(t) be a function that is sufficiently smooth and that decays sufficiently quickly near infinity so that its integrals exist. Further, let E(t) and E_nu be Fourier transform pairs so that

E(t)=int_(-infty)^inftyE_nue^(-2piinut)dnu
(1)
E^_(t)=int_(-infty)^inftyE^__(nu^')e^(2piinu^'t)dnu^',
(2)

where z^_ denotes the complex conjugate.

Then

int_(-infty)^infty|E(t)|^2dt=int_(-infty)^inftyE(t)E^_(t)dt
(3)
=int_(-infty)^infty[int_(-infty)^inftyE_nue^(-2piinut)dnuint_(-infty)^inftyE^__(nu^')e^(2piinu^'t)dnu^']dt
(4)
=int_(-infty)^inftyint_(-infty)^inftyint_(-infty)^inftyE_nuE^__(nu^')e^(2piit(nu^'-nu))dnudnu^'dt
(5)
=int_(-infty)^inftyint_(-infty)^inftyint_(-infty)^inftyE_nuE^__(nu^')e^(2piit(nu^'-nu))dtdnudnu^'
(6)
=int_(-infty)^inftyint_(-infty)^inftydelta(nu^'-nu)E_nuE^__(nu^')dnudnu^'
(7)
=int_(-infty)^inftyE_nuE^__nudnu
(8)
=int_(-infty)^infty|E_nu|^2dnu.
(9)

where delta(x-x_0) is the delta function.

See also

Fourier Transform, Parseval's Theorem, Power Spectrum

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References

Bracewell, R. "Rayleigh's Theorem." The Fourier Transform and Its Applications. New York: McGraw-Hill, pp. 112-113, 1965.Carleman, T. L'Intégrale de Fourier er questions qui s'y rattachent. Uppsala, Sweden: Almqvist and Wiksells, 1944.Rayleigh, J. W. S. "On the Character of Complete Radiation at a Given Temperature." Philos. Mag. 27, 1889. Reprinted in Scientific Papers. Cambridge, England: Cambridge University Press, 1899.Titchmarsh, E. C. "A Contribution to the Theory of Fourier Transforms." Proc. London Math. Soc. 23, 279, 1924.

Referenced on Wolfram|Alpha

Plancherel's Theorem

Cite this as:

Weisstein, Eric W. "Plancherel's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PlancherelsTheorem.html

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