Parseval's Theorem states that the sum of the squared magnitudes of a function's Fourier coefficients equals the integral of the squared magnitude of the function itself, so that Fourier decomposition preserves total energy. Named for Marc-Antoine Parseval, it is a basic identity of Fourier analysis with a direct analogue, Plancherel's theorem, for the Fourier transform on the real line.
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StatementParseval's theorem states that the Fourier transform is unitary, so the sum or integral of the squared magnitude of a function equals the sum or integral of the squared magnitude of its Fourier transform or Fourier coefficients, meaning Fourier decomposition preserves total energy. 2 Classification
Statement Form Parseval's Theorem
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Source Wolfram MathWorld: Parseval's Theorem
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1. Wikipedia: Parseval's theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.
2. Parseval's Theorem (Wikipedia)
Wikimedia Foundationlead paragraph, first sentence
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.
lead paragraph, second sentence naming the 1799 origin
It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series.
Proved By: Marc-Antoine Parseval, Lead paragraph
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the
View the Source Marc-Antoine Parseval (Wikipedia)
Principal Result Of: Marc-Antoine Parseval, Article textQuote, Principal Result Of: Marc-Antoine Parseval, Article text
Marc-Antoine Parseval des Chênes (27 April 1755-16 August 1836) was a French mathematician, most famous for what is now known as Parseval's theorem, which showed that the Fourier transform is unitary.
Wolfram MathWorld: Parseval's Theorem
Wolfram MathWorldIn Branch: Analysis, Subject classificationsQuote, In Branch: Analysis, Subject classifications
Calculus and Analysis Series
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