Proofs of Parseval’s Theorem & the Convolution Theorem

Proofs of parseval s Theorem & the Convolution Theorem (using the integral representation of the -function)1 The generalization of parseval s theoremThe result is f(t)g(t) dt=12 f( )g( ) d (1)This has many names but is often called Plancherel s key step in the proof of this is the use of the integral representation of the -function ( ) =12 e i d or ( ) =12 e i d .(2)We firstly invoke the inverse Fourier transformf(t) =12 f( )ei td (3)and then use this to re-write the LHS of (1) as f(t)g(t) dt= (12 f( )ei td )(12 g( 0) e i 0td 0)dt.(4)Re-arranging the order of integration we obtain f(t)f(t) dt=(12 )2 f( )g( 0) ( ei( 0)tdt) Use delta fn hered 0d .(5)The version of the integral representation of the -function we use in (2) above is ( 0) =12 eit( 0)dt.

Proofs of Parseval’s Theorem & the Convolution Theorem (using the integral representation of the δ-function) 1 The generalization of Parseval’s theorem The result is Z f(t)g(t)∗dt= 1 2π Z ∞ −∞ f(ω)g(ω)∗dω (1) This has many names but is often called Plancherel’s formula.

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  Theorem, Parseval s theorem, Parseval

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