We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1]$ with $f(0)=0$.
I am wondering if there exists an explicit construction of a sequence $f_{n} \in C_c^{\infty}(\mathbb R)$ such that
$$\lVert f-f_n\rVert_{C^{\alpha}([0,1])} \le \frac{1}{n}$$
and $\lvert f_n(x)\rvert \le \lvert f(x)\rvert$ on $[0,1]$. The usual convolution idea does not respect the last condition.
Best Answer
It is not possible: your condition $|f_n|<|f|$ implies that the zero set of $f_n$ is contained in the zero set of $f$. So $f(x)=|x|^{\alpha}$ cannot be approximated by a smooth function $f_n$, since $f_n(x)=(c+o(1))x$, and $\sup|f(x)-f_n(x)|\geq (1+o(x))|x|^\alpha$ for some small $|x|$.