Mollifier

Mollifiers are a tool for building smooth approximations of functions with sufficient regularity.

Misplaced &0 & \text{for} \space|x| \geq 1 \\ \end{cases}$$ is the **standard mollifier**, with $C$ chosen s.t. $\int_{\mathbb{R}^n} \eta(x) \space dx = 1$. Defining, $\eta_\varepsilon(x) = \frac1{\varepsilon^n} \eta(\frac{x}{\varepsilon})$ it holds for $f \in L_{\text{loc}}^p(U)$: $$ ||\eta_\varepsilon \ast f - f||_{L^p(V)} \to 0$$ as $\varepsilon \to 0$ for any $V \subset \overline{V} \subset U$, where

\eta_\varepsilon \ast f \in C^\infty(U_\varepsilon)

Paying in Attention