We assume that we have a $\alpha$-Holder 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

$$\Vert f-f_n\Vert_{C^{\alpha}([0,1])} \le \frac{1}{n}$$

and $\vert f_n(x)\vert \le \vert f(x)\vert$ on $[0,1].$ The usual convolution idea does not respect the last condition.