Application of Mollifier function.

Question

According to Wikipedia https://en.wikipedia.org/wiki/Mollifier, one of the uses of Mollifer functions is to smooth a function. How could you smooth with a mollifer function the function $f(x)=|x|$ at the origin (the "corner")? (I would like to be able to understand this illustrative example) Thank you.

Answer 1

you can observe behavior of $f_{\epsilon} := f * \Phi_{\epsilon}$ to see how it gets smoother. for any $\epsilon >0$, $f_{\epsilon}$ is $C^{\infty}$. I can give you some illustration for specific choose of $\Phi$ (Wikipedia example). (cause you mention you need illustrative example!, tell me if you need more technical details). you can see $f_{\epsilon} \xrightarrow[]{\epsilon \rightarrow 0} f$ (in some sence of convergence). more precisely since $f(x) = |x| \in L^p_{loc}(\mathbb{R})$ therefore $f_{\epsilon} \rightarrow f$ in $L^p_{loc}(\mathbb{R})$.