For any square-summable function the integral
converges in to some function as , i.e.
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).
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:
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
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).
|||A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)|
|||E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948)|
|||S. Bochner, "Lectures on Fourier integrals" , Princeton Univ. Press (1959) (Translated from German)|
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.
|[a1]||W. Rudin, "Fourier analysis on groups" , Wiley (1962)|
|[a2]||A. Weil, "l'Intégration dans les groupes topologiques et ses applications" , Hermann (1940)|
|[a3]||L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)|
|[a4]||E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1–2 , Springer (1979)|
|[a5]||H. Reiter, "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press (1968)|
Plancherel theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Plancherel_theorem&oldid=16182