= Orthonormal translates and Fourier periodization
{title2=$\sum_k|\widehat\phi(t+2\pi k)|^2=1$}
With $\widehat\phi(\xi)=\int\phi(x)e^{-ix\xi}\,dx$ in the <Plancherel theorem> sense, the integer translates of $\phi\in L^2(\mathbb R)$ are orthonormal exactly when
$$
\sum_{k\in\mathbb Z}|\widehat\phi(t+2\pi k)|^2=1\quad\text{almost everywhere}.
$$
The <Fourier coefficients> of the periodized energy are the translate inner products. <Uniqueness of Fourier coefficients in L1> therefore proves both directions. No absolute integrability assumption on $\phi$ is required.
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