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.



Approximation of Hölder continuous functions "from below"
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 $|x|$.


