= Cross-correlation
{title2=$(f\star g)(x)=\int_{\mathbb R}\overline{f(y)}g(x+y)\,dy$}
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> $\widehat{f\star g}(k)=\overline{\widehat f(k)}\widehat g(k)$. 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>.
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