Standard Mollifiers

Note

Consider the function

$$\rho (x)=\{\begin{array}{l}{e}^{-}\frac{1}{1-\Vert x{\Vert }^2}\text{ if }\Vert x\Vert <1\\ 0\text{ otherwise }\end{array}$$

The standard mollifiers are the family of functions $\{{\rho }_{ϵ}{\}}_{ϵ>0}$

$${\rho }_{ϵ}(x):=c_{ϵ}\rho (ϵx)$$

where $c_{ϵ}$ is a normalization factor.

The standard Mollifiers are an example of Test functions with the following nice (usefull) properties:

• $\Vert {\rho }_{ϵ}{\Vert }_{L^1}=1$ by definition.
• ${\rho }_{ϵ}\in C_c^{\infty }$
• ${\lim }_{ϵ\to 0}{\rho }_{ϵ}(x)={\delta }_0$ in the sense of Distributions.
• Smoothing property: given any distribution $T$, the family of convolutions

$$T_{ϵ}=T\ast {\rho }_{ϵ}$$

are smooth functions.

A nice smooth bump! 👌