According to, say, Wikipedia, the Poisson Summation Formula states that $$\sum_{n=-\infty}^\infty s(n) = \sum_{k=-\infty}^\infty S(k),$$ where $$S(k) = \int_{-\infty}^\infty s(x) e^{-2\pi i k x}dx.$$ Now from, say, the DLMF, we also have the formal series representation of the Dirac delta function, $$\delta(x-a) = \frac{1}{2\pi}\sum_{k=-\infty}^\infty e^{ik(x-a)}.$$ Combining the above, we should then have that $$\sum_{n=-\infty}^\infty s(n) = \sum_{k=-\infty}^\infty S(k) \\ = \sum_{k=-\infty}^\infty\int_{-\infty}^\infty s(x) e^{-2\pi i k x}dx \\ = \int_{-\infty}^\infty s(x)\left( \sum_{k=-\infty}^\infty e^{-2\pi i k x} \right) dx \\ = \int_{-\infty}^\infty s(x)\delta(x) dx \\ = s(0).$$
$\sum_{n=-\infty}^\infty s(n) = s(0)$ cannot be correct in general. So what went wrong?
Presumably the swapping of the sum and the integral is the source of the issue. If so, my question, then, is: when is the swapping of the sum and the integral in this case valid?
Further, clearly $\sum_{n=-\infty}^\infty s(n) = s(0)$ when $s(x)$ is odd, $s(-n)=-s(n)$; are there more general conditions for $s(x)$ such that $\sum_{n=-\infty}^\infty s(n) = s(0)$?
Thank you very much for your insight!