Cross-correlation ($(f\star g)(x)={\int }_{\mathbb{R}}\overline{f(y)}g(x+y)dy$)

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 $\widehat{f\star g}(k)=\overline{\hat{f}(k)}\hat{g}(k)$. 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.

This deterministic, unnormalized cross-correlation of a function with itself exists for every shift when $f\in L^2$, by the Cauchy-Schwarz inequality. It satisfies $h(0)=\Vert f{\Vert }_2^2$ and $h(-x)=\overline{h(x)}$. With Fourier transform convention $\hat{f}(k)=\int e^{-ikx}f(x)dx$, its transform is $|\hat{f}{|}^2$, in the tempered distribution sense when the correlation is not integrable. Fourier inversion at zero gives the Parseval identity, first for Schwartz functions and then by approximation in L2 space. Unlike the normalized statistical autocorrelation, its value at zero is not generally one.