# Plancherel theorem

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For any square-summable function the integral converges in to some function as , i.e. (1)

Here the function itself is representable as the limit in of the integrals as , i.e. Also, the following relation holds: (the Parseval–Plancherel formula).

The function where the limit is understood in the sense of convergence in (as in (1)), is called the Fourier transform of ; it is sometimes denoted by the symbolic formula: (2)

where the integral in (2) must be understood in the sense of the principal value at in the metric of . One similarly interprets the equation (3)

For functions , the integrals (2) and (3) exist in the sense of the principal value for almost all .

The functions and also satisfy the following equations for almost-all :  If Fourier transformation is denoted by and if denotes the inverse, then Plancherel's theorem can be rephrased as follows: and are mutually-inverse unitary operators on (cf. Unitary operator).

The theorem was established by M. Plancherel (1910).

How to Cite This Entry:
Plancherel theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Plancherel_theorem&oldid=16182
This article was adapted from an original article by P.I. Lizorkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article