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Plancherel theorem
Theorem in harmonic analysis
Plancherel theorem
In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It is a generalization of Parseval's theorem; often used in the fields of science and engineering, proving the unitarity of the Fourier transform. 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. That is, if f ( x ) {\displaystyle f(x)} is a function on the real line, and f ^ ( ξ ) {\displaystyle {\widehat {f}}(\xi )} is its frequency spectrum, then
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