Cross-correlation

For this sign convention, cross-correlation compares a function with translated copies of another, conjugating the first factor. If $f,g\in L^1(\mathbb{R})$, Fubini's theorem gives Fourier transform $\widehat{f\star g}(k)=\overline{\hat{f}(k)}\hat{g}(k)$. Indeed, the substitution $s=x+y$ factors the double integral into $\int \overline{f(y)}{e}^{iky}dy$ and $\int g(s)e^{-iks}ds$. The zero-lag value for $f=g\in L^2$ is the squared L2 norm. Fourier inversion applied to this correlation proves the Parseval identity, first for Schwartz functions and then by density for L2 space.