Parseval identity

For $f,g\in L^2({\mathbb{R}}^n)$ and the convention $\hat{f}(\xi )=\int f(x)e^{-ix\cdot \xi }dx$,

For a square-integrable function of period $T$, use Fourier coefficients $u_r=T^{-1}{\int }_0^{T}f(t)e^{-2\pi irt/T}dt$. Fourier orthogonality proves the displayed identity for a finite trigonometric polynomial. The mean-square convergence of the Fourier series then passes it to the limit. It includes the zero-frequency contribution $|u_0{|}^2$; excluding that contribution gives the variance rather than the full mean-square signal power.

Applying the Parseval identity to the repeated exponential convolution gives