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

The heart of Plancherel's theorem is the assertion that if , then: a) , where is defined by (2) for ; b) ; and c) the set of all such is dense in . Then one extends this mapping to a unitary mapping of onto itself which satisfies for almost every . There are generalizations of Plancherel's theorem in which is replaced by or by any locally compact Abelian group. Cf. also Harmonic analysis, abstract.

References