Plancherel's theorem states that the integral of the squared complex modulus of a Schwartz function is equal to that of its Fourier transform in the normalization below. 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. Plancherel (1910) first established conditions under which the theorem holds (Titchmarsh 1924; Bracewell 1965, p. 113).
Let 
be a Schwartz function, and let 
and 
be Fourier transform
pairs so that
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
denotes the complex conjugate.
A formal calculation using the delta function gives
 |  |  |
(3)
|
 |  | ![int_(-infty)^infty[int_(-infty)^inftyE_nue^(-2piinut)dnuint_(-infty)^inftyE^__(nu^')e^(2piinu^'t)dnu^']dt](/images/equations/PlancherelsTheorem/Inline16.svg) |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
where 
is the delta function. The integration in 
that produces the delta
function is interpreted in the sense of distributions,
rather than as an interchange of absolutely convergent triple integrals
(Brown 2001, pp. 17-21).
