First of all this has nothing to do with the inflection point of $e^{-\alpha x^2}$. According to Poisson summation formula (see Whittaker, Watson, Modern analysis, chapter 21.51)
$$
\sum_{n=-\infty}^\infty e^{-\alpha (x-n)^2}=2{\sqrt{\frac{\pi}{\alpha}}}\left(1+2\sum_{n=1}^\infty e^{-\frac{\pi^2}{\alpha}n^2}\cos2\pi n x\right),\quad \text{Re}~\alpha>0.\tag{1}
$$
From this formula one can see that when $\alpha>0$ is small, the Fourier coefficients $a_n=2{\sqrt{\frac{\pi}{\alpha}}}e^{-\frac{\pi^2}{\alpha}n^2}$ of the resulting function decreases rapidly with increasing $n$. When $\alpha=1/2$ one has
$$
\frac{a_1}{a_0}=2e^{-2\pi^2}\approx 5.4\times 10^{-9}.
$$

I don't know any conceptual explanation for this in mathematics, but there is such an explanation that comes from physics. Consider a quantum particle with mass $m$ on a ring of radius $a$. We assume the ring is pierced with magnetic flux $\phi$. We want to calculate partition function of this system at temperature $T>0$ in two different ways.

On the one hand, it is known that the energy spectrum of the particle is given by
$$
E_n=\frac{\hbar^2}{2m^2}\left(n+\frac{\phi}{\phi_0}\right)^2,
$$
where $\phi_0$ is the so called magnetic flux quantum. Then partition function is given as the Gibbs sum
$$
Z=\sum_{n=-\infty}^\infty e^{-E_n/T}=\sum_{n=-\infty}^\infty e^{-\frac{\hbar^2}{2m^2T}\left(n+{\phi}/{\phi_0}\right)^2}.\tag{2}
$$
Here one immediately recognizes the sum analogous to the LHS of $(1)$.

On the other hand, partition function is related to the trace of the density matrix $\rho_{\theta_1,\theta_2}$, i.e. to $\int\rho_{\theta,\theta}d\theta$. It is possible to calculate the density matrix of this system in imaginary time representation by solving a certain differential equation. Details of this calculation can be found for example in this book, chapter 4.3. Consider first the more simple case of unbounded line $-\infty<\theta<+\infty$; then the answer is
$$
\rho_{\theta_1,\theta_2}=Ce^{-\frac{mTa^2(\theta_1-\theta_2)^2}{2\hbar^2}},
$$
where $C$ is some normalization constant. Magnetic flux does not enter this expression because there is not any nontrivial loop in the system. On a ring there is a nontrivial loop, and let $n$ be the winding number. It is known that the partition function is a sum over all homotopy classes multiplied by corresponding phase factor. In this case the phase factor comes from the magnetic flux (Aharonov-Bohm phase) and equals $e^{2\pi in\phi/\phi_0}$.
As a result we have
\begin{align}
Z&=\sum_{n=-\infty}^\infty e^{2\pi in\phi/\phi_0}\int\rho_{\theta,\theta+2\pi n}d\theta\\
&=2\pi C \sum_{n=-\infty}^\infty e^{-\frac{2\pi^2mTa^2}{\hbar^2}n^2+2\pi in\phi/\phi_0}.\tag{3}
\end{align}
Combining $(2)$ and $(3)$ one obtains the transformation in $(1)$.

One can see from this physical interpretation that when the temperature $T$ increases, then the higher Fourier harmonics decrease. This corresponds to the physical intuition that the higher the temperature the more chaotic the system becomes and the effect of the Aharonov-Bohm phase averages out due to thermal fluctuations.

smoothing parameterof a lattice in lattice-based cryptography. $\endgroup$2more comments