3
$\begingroup$

I have a question about approximation by smooth functions.

Let $\Omega$ be an open subset of $\mathbb{R}^{d}$ and $\bar{\Omega}$ be its closure in $\mathbb{R}^{d}$.

$C_{c}(\bar{\Omega})$ denotes the space of all continuous real valued functions on $\bar{\Omega}$ with compact support. i.e. If $f \in C_{c}(\bar{\Omega})$ then "$\text{supp}[f]:= $the closire of $\left\{x \in \bar{\Omega} : f(x) \neq 0 \right\}$ in $\bar{\Omega}$" is compact subset of $\bar{\Omega}$.

Take $f \in C_{c}(\bar{\Omega})$. By Tietze extension theorem, there exists $F \in C_{c}(\mathbb{R}^{d})$ such that $F=f$ on $\bar{\Omega}$. Define $F_{\delta}(x):=\int_{\mathbb{R}^{d}}j_{\delta}(x-y)F(y)\,dy$, where $j_{\delta}$ is standard mollifier. By the definition of standard mollifier, we can see \begin{equation*} F_{\delta}=0 \text{ on } \mathbb{R}^{d} \setminus \left\{x=x_{1}+x_{2} : x_{1} \in \text{supp}[f], |x_{2}| \leq \delta \right\}. \end{equation*} i.e. $\text{supp}[f] \subset \text{supp}[F_{\delta}]$ Moreover, we can deduce $\sup_{x \in supp[f]} \left|F_{\delta}(x)-f(x) \right| \to 0$ as $\delta \to 0$.

My question

Can we find $G_{\delta} \in C_{c}^{\infty}(\mathbb{R}^{d})$ s.t. $\text{supp}[G_{\delta}] \subset \text{supp}[f] $ and $\sup_{x \in supp[f]} \left|G_{\delta}(x)-f(x) \right| \to 0$ as $\delta \to 0$ ?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.