How to prove the Parseval's Formula?

Question

How to prove Parseval's Formula: $$ \int_{\mathbb{R}^{n}} \phi(x) \bar{\psi}(x) dx = (2\pi)^{-n}\int_{\mathbb{R}^{n}}\hat{\phi}(\xi)\bar{\hat{\psi}}(\xi) d \xi, $$ where $ \phi(x), \xi(x)\in L^{2}(\mathbb{R}^{n}) $. And for a function $ f(x)\in L^{1}(\mathbb{R^n}) $, define its Fourier transform to be $$ \hat{f}(\xi)=\int_{\mathbb{R}^{n}}e^{-i<x, \xi>}f(x)dx. $$ And $ \bar{\psi}(x) $ represents the complex conjugate of function $ \psi(x) $

Answer 1

I will adopt the following convention for the Fourier transform (i.e. no factors of $2\pi$ for the back transformation):

$$\phi(\xi)=(2\pi)^{n}\int_{\mathbb{R}^{n}}\hat{\phi}(x)e^{-i x \xi}\, dx$$

Then $$\langle\phi,\psi\rangle=\int_{\mathbb{R}^{n}} \phi(x) \bar{\psi}(x) dx=\int_{\mathbb{R}^{n}} \phi(x) \left(\overline{\int_{\mathbb{R}^{n}}\hat{\psi}(\xi)e^{i x \xi}\, d\xi} \right)dx=\\ =\int_{\mathbb{R}^{n}} \phi(x) \int_{\mathbb{R}^{n}}\bar{\hat{\psi}}(\xi)e^{-i x \xi}\, d\xi \,dx=\int_{\mathbb{R}^{n}} \left(\int_{\mathbb{R}^{n}}\phi(x)e^{-i x \xi}\, dx \right)\bar{\hat{\psi}}(\xi)d\xi\\ =(2\pi)^{-n}\int_{\mathbb{R}^{n}}\hat{\phi}(\xi)\bar{\hat{\psi}}(\xi) d \xi=(2\pi)^{-n}\langle\hat{\phi},\hat{\psi}\rangle$$.

This would typically be referred to as the Plancharel theorem, rather than Parseval's formula.