Gaussian approximate identity
= Gaussian approximate identity
The centered normal densities
$$
g_t(x)=\frac1{\sqrt{2\pi t}}e^{-x^2/(2t)}
$$
form an <approximate identity>: for every $f\in L^1(\mathbb R)$, $f*g_t\to f$ at almost every point where the <Lebesgue differentiation theorem> applies. Their Fourier transforms are $e^{-t\xi^2/2}$ in the angular-frequency convention.