Parseval's theorem

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<div class="definition"><div class="short_definition">A theorem relating the product of two functions to the products of their [[Fourier  series]] components.</div><br/> <div class="paragraph">If the functions are ''f''(''x'') and ''F''(''x''), and their Fourier series components have respective amplitudes  ''a''<sub>''n''</sub>, ''b''<sub>''n''</sub> and ''A''<sub>''n''</sub>, ''B''<sub>''n''</sub>, Parseval's theorem states that under certain general conditions  <div class="display-formula"><blockquote>[[File:ams2001glos-Pe3.gif|link=|center|ams2001glos-Pe3]]</blockquote></div> There is an analogous theorem for [[Fourier transforms]].</div><br/> </div>
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<div class="definition"><div class="short_definition">A theorem relating the product of two functions to the products of their [[Fourier series|Fourier  series]] components.</div><br/> <div class="paragraph">If the functions are ''f''(''x'') and ''F''(''x''), and their Fourier series components have respective amplitudes  ''a''<sub>''n''</sub>, ''b''<sub>''n''</sub> and ''A''<sub>''n''</sub>, ''B''<sub>''n''</sub>, Parseval's theorem states that under certain general conditions  <div class="display-formula"><blockquote>[[File:ams2001glos-Pe3.gif|link=|center|ams2001glos-Pe3]]</blockquote></div> There is an analogous theorem for [[Fourier transforms]].</div><br/> </div>
 
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Latest revision as of 19:34, 25 April 2012


Parseval's theorem

A theorem relating the product of two functions to the products of their Fourier series components.

If the functions are f(x) and F(x), and their Fourier series components have respective amplitudes an, bn and An, Bn, Parseval's theorem states that under certain general conditions
ams2001glos-Pe3
There is an analogous theorem for Fourier transforms.

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