Let be $u:\mathbb{R}\rightarrow\mathbb{R}$ a convex function. I consider $u_n=u*\rho_n$, defined on $\mathbb{R}$, for $n\in\mathbb{N}$,
where $\rho_\epsilon(x)=\epsilon^{-n}\rho(\dfrac{x}{n})$ (the standard mollifier) and $$\rho(x) = \begin{cases} e^{1/(1-|x|^2)}/I& \text{ if } |x| < 1\\ 0& \text{ if } |x|\geq 1 \end{cases},$$ ( $I$ is a coefficient of normalization).
Let be $R>0$. I know that the sequence $(u_n)$ converges uniformly to $u$ in $[0,R]$.
- Is it true that the sequence of the derivative $(u'_n)$ converges almost everywhere to $u'$ in $[0,R]$? I think that the answer is yes, but I can't find a reference.
- Can I say something about the convergenge of $(u''_n)$? Is it true that $(u''_n)$ converges almost everywhere to $u''$?
Can someone help me? Thanks