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.
Facts
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
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Source Plancherel theorem (Wikipedia)
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Source Plancherel theorem (Wikipedia)
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1. Plancherel theorem (Wikipedia)
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.
Lead paragraph
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
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