= Orthogonality of integer translates
{title2=$\langle f(\cdot+k),f(\cdot+l)\rangle=(2\pi)^{-1}\int|\widehat f(t)|^2e^{i(k-l)t}\,dt$}
With the Fourier convention $\widehat f(t)=\int f(x)e^{-itx}\,dx$, translation by $k$ multiplies the transform by $e^{ikt}$. The <Plancherel theorem> gives the displayed correlation identity for every square-integrable function, by extension from integrable square-integrable functions. Thus distinct integer translates are orthogonal exactly when the nonzero integer Fourier <integrals> of $|\widehat f|^2$ vanish. The products in both <integrals> are integrable by the <Cauchy-Schwarz inequality>.
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