How to choose function $\sum_{m\in \mathbb Z} (-1)^m f(x+m) f(x-m+n)=0$?

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Can we expect to choose a function $f:\mathbb R \to \mathbb R$ (nonzero compactly supported) so that

$\sum_{m\in \mathbb Z} (-1)^m f(x+m) f(x-m+n)=0$ for all $x\in \mathbb R$ and $n\in \mathbb Z$?