Parseval's theorem and Fourier Transform proof

  • #1

Summary:

I am searching for a proof that the square of a function is equal to the sum of the square of its transform.
Given a function F(t)
$$ F(t) = \int_{-\infty}^{\infty} C(\omega)cos(\omega t) d \omega + \int_{-\infty}^{\infty} S(\omega)sin(\omega t) d \omega $$
I am looking for a proof of the following:

$$ \int_{-\infty}^{\infty} F^{2}(t) dt= 2\pi\int_{-\infty}^{\infty} (C^{2}(\omega) + S^{2}(\omega)) d \omega $$
 
Last edited:

Answers and Replies

  • #5
Infrared
Science Advisor
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The idea is similar. Parseval's identity says that taking Fourier series gives an isometry ##L^2(S^1)\to\ell^2##. Plancherel says that the Fourier transform gives a self-isometry of ##L^2(\mathbb{R})\cap L^1(\mathbb{R})##.
 

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