Let $u \in AC([a,b])$ with $u' \in L^2(a,b) $ and $u(a)=u(b)=0$. Let $(\rho_n)_n$ be a sequence of mollifiers in $\mathbb{R}$ and $\overline{u'} : \mathbb{R} \mapsto \mathbb{R}$ defined by $\overline{u'}=u'$ in $(a,b)$, $\overline{u'}=0$ otherwise. Let $v_n,u_n : \mathbb{R} \mapsto \mathbb{R}$ be the sequences defined as follows: $$ v_n(x):=(\rho_n * \overline{u'})(x) \qquad \text{and} \qquad u_n(x):=\int_{a}^{x} (v_n(t)+c_n) \, dt, \ \ x \in \mathbb{R} $$, where $$ c_n:=-\frac{\int_{a}^{b} v_n(t) \, dt}{b-a} \ \ \forall n \in \mathbb{N}. $$ Prove that $(u_n)_n \subset \{ v \in C^2([a,b]) : v(a)=v(b)=0 \}$, $u_n \rightarrow u $ and $u_n' \rightarrow u'$ in $L^2(a,b)$. I'm fine with the first question, but when trying to prove the two convergence statements I'm immediately stuck dealing with those $L^2$ norms. Maybe there are some useful properties of convolutions that I'm ignoring (or a theorem that avoids heavy calculations)? Any hint is appreciated.
Convergence of a sequence in $L^2$, mollifiers
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real-analysis
functional-analysis
lp-spaces
convolution
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1A mollifier is continuous with respect to $L^p$ for any $1 \le p < \infty$. That is, $\|\rho_n * f\|_{L^p} \le \|f\|_{L^p}$. This is proved using Holder's inequality after writing $\rho = \rho^{1/p}\rho^{1/q}$. The case $p=1$ is handled separately and easily. – 2017-01-16