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is there exist the alternating analogue to Poisson sum formula ??

i mean

$$ \sum_{m=-\infty}^{\infty} (-1)^{m}f(m) = \sum_{m=-\infty}^{\infty}F(2\pi m) $$

here $ F(u)= \int_{-\infty}^{\infty} dx f(x)e^{iux} $

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