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Plancherel theorem - Wikipedia

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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. 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 ) is a function on the real line, and 𝑓 ^ ( 𝜉 ) is its frequency spectrum, then ∫ − ∞ ∞ | 𝑓 ( 𝑥 ) | 2 𝑑 𝑥 = ∫ − ∞ ∞ | 𝑓 ^ ( 𝜉 ) | 2 𝑑 𝜉 The Fourier transform of an L1 function 𝑓 on the real line 𝑅 is defined as the Lebesgue integral 𝑓 ^ ( 𝜉 ) = ∫ 𝑅 𝑓 ( 𝑥 ) 𝑒 − 2 𝜋 𝑖 𝑥 𝜉 𝑑 𝑥 . If 𝑓 belongs to both 𝐿 1 and 𝐿 2 , then the Plancherel theorem states that 𝑓 ^ also belongs to 𝐿 2 , and the Fourier transform is an isometry with respect to the L2 norm, which is to say that ∫ − ∞ ∞ | 𝑓

Plancherel theorem - Wikipedia Jump to content From Wikipedia, the free encyclopedia Theorem in harmonic analysis 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 ) {\displ

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