We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1].$

I am wondering if there exists an explicit construction of a sequence $f_{n} \in C_c^{\infty}(\mathbb R)$ such that

$$\lVert f-f_n\rVert_{C^{\beta}([0,1])} \le \frac{1}{n}$$

for fixed $\beta<\alpha$ and $\lvert f_n(x)\rvert \le \lvert f(x)\rvert$ on $[0,1]$. The usual convolution idea does not respect the last condition. In an earlier post, I mistakingly took $\beta=\alpha.$