Finding minima and maxima of the map $f(x) = \sum_{k \in \mathbb Z} e^{-c(x+k)^2}, x \in [0,1]$ Suppose that $c>0$ and define $f : [0,1] \to \mathbb R,$
$$
f(x) = \sum_{k \in \mathbb Z} e^{-c(x+k)^2}
$$
I want to show that $x=1/2$ is a minimizer of this map and the maxima are attained at $x=0$ and $x=1$. The above map seems to be a Jacobi theta function of third kind. I'm not very familiar with special functions so any help or reference would be appreciated!

As you stated correctly, the function $f$ is essentially Jacobi's theta function of third kind, defined via
$$
\vartheta_3(z,q) = \sum_n q^{n^2}e^{2inz}.
$$
Suppose that $q \in (0,1)$. An application of Poisson's summation formula shows that for a suitable $q$, the map $f$ in your question coincides with $\vartheta_3$. The fact that $f$ attains its maximum at $x \in \{ 0,1 \}$ and its minimum at $x=\frac 1 2$ follows from the result that $\vartheta_3$ (considered as a $\pi$-periodic map) has minima at $\pi(k+\frac 1 2)$ and maxima at $\pi k$ where $k$ is an arbitrary integer. You can find this, for instance, in the paper by Janssen (page 178).
Janssen, A. J. E. M., Some Weyl-Heisenberg frame bound calculations., Indag. Math., New Ser. 7, No. 2, 165-183 (1996). ZBL1056.42512.
