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For a real function $f$ defined on the reals, and real $y$, define the 2-way infinite sum $$ F_f(y) = \sum_{i\in\mathbb Z} f(y+i). $$

If $F_f(y)$ is defined for all $y$, it is periodic of period 1.

Is there a characterisations of the class of smooth functions $f$ such that $F_f(y)=0$ for all $y$? I don't know what definition of "smooth" provides the most satisfactory answer, so I'll leave it for you to choose. Two initial observations are that the class is closed under addition, under multiplication by a smooth function of period 1, and under the operation $f(x)\mapsto f(x/k+\ell)$ for integers $k,\ell$.

Also, is there some systematic study of the transform $f\mapsto F_f$?

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    $\begingroup$ Does this Wikipedia page help you ? $\endgroup$
    – Hachino
    Commented Feb 18, 2015 at 12:55
  • $\begingroup$ By the Poisson summation formula, the expected answer would be the condition $\widehat{f}(n)=0$ for all $n\in\mathbb Z$. Are you looking for something else? $\endgroup$
    – Christian Remling
    Commented Feb 18, 2015 at 17:47
  • $\begingroup$ Hachino and Christian: Thanks, I was sure I was missing something I'd seen before. $\endgroup$
    – Brendan McKay
    Commented Feb 19, 2015 at 3:25

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