Let $\eta \in \mathcal{C}^{\infty}_{0}(\mathbb{R})$, where $\mathcal{C}^{\infty}_{0}(\mathbb{R})$ is the set of compactly supported infinitely differentiable function, be a function which is
- non-negative, non-vanishing and constant in a neighborhood of $x=0$,
- symmetric about $x=0$,
- supported in $[-1,1]$,
- bounded between $0$ and $\eta(0)$, and
- normalized so that $\int \eta d\mu =1$.
Let $\varphi$ be a compactly supported absolutely continuous function with a piecewise continuous derivative $\varphi'$. I have been told (by a user of MSE whom I thank very much again) the following, which I think to be interesting enough to be the object of a specifical question:
Then one defines$$\eta_{n}(x) = n\eta(x/n)$$so that $\int_{\mathbb{R}}\eta_{n}\,d\mu =1$ for all $n$. For any compactly supported absolutely continuous function $\varphi$, define$$\varphi_{n}=\int_{\mathbb{R}}\eta_{n}(x-y)\varphi(y)\,d\mu(y).$$The function $\varphi_{n}$ is in $\mathcal{C}^{\infty}_{0}(\mathbb{R})$. Because $\eta_{n}$ is supported in $[-1/n,1/n]$, and $\varphi$ is continuous, then $\varphi_{n}$ converges uniformly to $\varphi$ as $n\rightarrow\infty$. And, because $\varphi$ is absolutely continuous, then$$\varphi_{n}'=\int_{\mathbb{R}}\eta_{n}'(x-y)\varphi(y)\,d\mu(y)=-\int_{\mathbb{R}}\eta_{n}(x-y)\varphi'(y)\,d\mu(y).$$ For my case $\varphi'$ is piecewise continuous, and the right side then converges pointwise everywhere to mean of the left- and right-hand limits of $\varphi'$, and it remains uniformly bounded by any bound for $\varphi'$.
I must say that I have no knowledge of the theory of mollification until now.
I would like to understand why
- the fact that $\eta_{n}$ is supported in $[-1/n,1/n]$, and $\varphi$ is continuous, implies that $\varphi_{n}$ converges uniformly to $\varphi$ as $n\rightarrow\infty$;
- the absolute continuity of $\varphi$ implies $\varphi_{n}'=\int_{\mathbb{R}}\eta_{n}'(x-y)\varphi(y)\,d\mu(y)=-\int_{\mathbb{R}}\eta_{n}(x-y)\varphi'(y)\,d\mu(y)$;
- the piecewise continuity of $\varphi'$ implies the facts that $-\int_{\mathbb{R}}\eta_{n}(x-y)\varphi'(y)\,d\mu(y)$ converges to $(\lim_{t\to x^+}\varphi'(t)+\lim_{t\to x^-}\varphi'(t))/2$ and $|\int_{\mathbb{R}}\eta_{n}(x-y)\varphi'(y)\,d\mu(y)|\le |\varphi'(x)|$.
I heartily thank both who told me these interesting facts and whoever will help me to understand the reason of the quoted facts.