Plancherel's Theorem

$$\int_{-\infty}^\infty f(x)g^*(x)\,dx = \int_{-\infty}^\infty F(s)G^*(s)\,ds,$$

where $F(s)\equiv {\mathcal F}[f(x)]$ and ${\mathcal F}$ denotes a Fourier Transform. If $f$ and $g$ are real

$$\int_{-\infty}^\infty f(x)g(-x)\,dx = \int_{-\infty}^\infty F(s)G(s)\,ds.$$

See also Fourier Transform, Parseval's Theorem