# 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> | + | <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 There is an analogous theorem for Fourier transforms.

*f*(*x*) and*F*(*x*), and their Fourier series components have respective amplitudes*a*_{n},*b*_{n}and*A*_{n},*B*_{n}, Parseval's theorem states that under certain general conditions