Mollifier

Remarks

Each $\varphi_\varepsilon$ is again smooth and nonnegative, has integral $1$ by the change of variables $x = \varepsilon y$, and has support in the ball of radius $\varepsilon$. Some authors drop the sign condition 3 or replace condition 2 by compact support anywhere; the approximation properties remain.

Example: the standard mollifier is $$\varphi(x) = \begin{cases} c \, \exp\left( \dfrac{-1}{1 - \abs{x}^2} \right), & \abs{x} < 1, \\ 0, & \abs{x} \ge 1, \end{cases}$$ with $c > 0$ chosen so that the integral is $1$. It is positive inside the ball and vanishes outside, so conditions 2 and 3 hold. It is smooth because $t \mapsto e^{-1/t}$ for $t > 0$, extended by $0$ for $t \le 0$, is smooth, and $\varphi(x) = c \, e^{-1/t}$ with $t = 1 - \abs{x}^2$.

Non-example: the Gaussian $\varphi(x) = \pi^{-n/2} e^{-\abs{x}^2}$ is smooth, positive and has integral $1$. Its support is all of $\R^n$, so it fails condition 1 and condition 2. It still gives an approximate identity, but convolution with it does not preserve compact support.