Gaussian function

A Gaussian function has the form $A\exp [-a(x-b{)}^2]$ with $a>0$. Its integral and moments reduce by translation and scaling to the Gaussian integral.

For fixed $c,h>0$, $L\ge 1$ and $\epsilon \ge 0$, complete the square in the exponent. A translated Gaussian function summed on a fixed-spaced lattice has mass $O(\sqrt{L})$ uniformly in the translate, by comparison with its integral and its maximum. This keeps a square-root logarithm, rather than the full number of dyadic intervals, when summing exponentially damped exponential sum bounds.