Density of smooth positive functions

Question

Let $\Omega$ be an open bounded set of $R^n$. For $f\in L^2(\Omega)$ such that $f>0$, a.e. in $\Omega, $ there is $(f_k)\subset W^{2,\infty}(\Omega)$ such that $f_k\to f$ in $L^2(\Omega)$. My question is:

Is it possible to chose $f_k>0,\; a.e. \; \Omega, \forall k?$

Answer 1

Yes. This can be obtained by the typical approach via mollification:

• Extend $f$ to $\mathbb{R}^n \setminus \Omega$ by $1$.
• Let $f_k$ be the convolution of $f$ with a smooth convolution kernel.
• This directly yields $f_k > 0$ a.e. and $f_k \to f$ in $L^2(\Omega)$ if the convolution kernels are appropriately chosen.