In mathematics, the Plancherel theorem is a result in harmonic analysis, first proved by Michel Plancherel. In its simplest form it states that if a function f is in both L¹(R) and L²(R), then its Fourier transform is in L²(R); moreover the Fourier transform map is isometric. This implies that the Fourier transform map restricted to L¹(R) ∩ L²(R) has a unique extension to a linear isometric map L²(R) →L²(R). This isometry is actually a unitary map. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... Michel Plancherel (1885-1967) was a Swiss mathematician. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ... In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ...

Here Plancherel's version concerns spaces of functions on the real line. The theorem is valid in abstract versions, on locally compact abelian groups in general. Even more generally, there is a version of the Plancherel theorem which makes sense for non-commutative locally compact groups satisfying certain technical assumptions. This is the subject of non-commutative harmonic analysis. In mathematics, the real line is simply the set of real numbers. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ...

The unitarity of the Fourier transform is often called Parseval's theorem in science and engineering fields, based on an earlier (but less general) result that was used to prove the unitarity of the Fourier series. In mathematics, Parsevals theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. ... In mathematics, a Fourier series of a periodic function represents the function as a sum of periodic functions of the form where e is Eulers number and i the imaginary unit. ...

References

• J. Dixmier, Les C*-algèbres et leurs Represéntations, Gauthier Villars, 1969
• K. Yosida, Functional Analysis, Springer Verlag, 1968