Let $\phi \in L^2(\mathbb{R})$ and $\hat{\phi}$ be the Fourier transform of $\phi.$
Does this mean that $\sum_{m \in \mathbb{Z}} |\hat{\phi}(x + 2 \pi m)|^2$ converges in the $L^1$ sense on each compact set to a $2 \pi $ periodic function?
2026-04-19 21:31:36.1776634296
Fourier transform and $L^1,$ $L^2$ convergence
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