Plancherel theorem

From Academic Kids

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.

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.

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.

References

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