Orthogonality of integer translates ($⟨f(\cdot +k),f(\cdot +l)⟩=(2\pi {)}^{-1}\int |\hat{f}(t){|}^2{e}^{i(k-l)t}dt$)

With the Fourier convention $\hat{f}(t)=\int f(x)e^{-itx}dx$, 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 $|\hat{f}{|}^2$ vanish. The products in both integrals are integrable by the Cauchy-Schwarz inequality.