The Plancherel Theorem is a result of harmonic analysis, proved by Michel Plancherel in 1910, establishing that the Fourier transform is a unitary operation on the real line, meaning it preserves the total energy, or L2 norm, of a function. It generalizes Parseval's Theorem, which states the analogous energy-preserving property for Fourier series, to the Fourier transform, and its unitarity is a large part of why the Fourier transform is used so widely throughout science and engineering.
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StatementThe integral of the squared modulus of a function equals the integral of the squared modulus of its Fourier transform, meaning the Fourier transform preserves a function's total energy. 1 Classification
Statement Form Plancherel Theorem
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Source Plancherel theorem (Wikipedia)
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Source MacTutor History of Mathematics, University of St Andrews: Michel Plancherel
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Introduction section
The theorem states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum.
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In mathematics, the Plancherel theorem (sometimes called the Parseval-Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910.
- In Branch: Harmonic Analysis, Lead sentence
Proved By: Michel Plancherel, Lead sentence, attribution
proven by Michel Plancherel in 1910
View the SourceMacTutor History of Mathematics, University of St Andrews: Michel Plancherel
MacTutor History of Mathematics, University of St AndrewsPrincipal Result Of: Michel Plancherel, Biography textQuote, Principal Result Of: Michel Plancherel, Biography text
He is best known for Plancherel's theorem in harmonic analysis.
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