# Plancherel theorem

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).

#### References

 [1] A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988) [2] E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948) [3] S. Bochner, "Lectures on Fourier integrals" , Princeton Univ. Press (1959) (Translated from German)