= Autocorrelation of a square-integrable function
{title2=$h(x)=\int_{\mathbb R}\overline{f(y)}f(y+x)\,dy$}
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 $h(0)=\|f\|_2^2$ and $h(-x)=\overline{h(x)}$. With <Fourier transform> convention $\widehat f(k)=\int e^{-ikx}f(x)dx$, its transform is $|\widehat f|^2$, 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.
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