Undergrad Parseval's theorem and Fourier Transform proof

[Ineedhelp0]

TL;DR

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:

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

 

[fresh_42]

Have a look at https://www.physicsforums.com/threads/math-challenge-november-2019.979872/#post-6256447

 

[Infrared]

This is known as Plancherel's theorem. If you're okay with the use of the Dirac delta, here is an argument http://mathworld.wolfram.com/PlancherelsTheorem.html

 

[fresh_42]

This is known as Plancherel's theorem. If you're okay with the use of the Dirac delta, here is an argument http://mathworld.wolfram.com/PlancherelsTheorem.html

Looks like the same idea.

 

[Infrared]

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 $L^2(\mathbb{R})\cap L^1(\mathbb{R})$.