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grammar
Jukka Kohonen
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Approximation of Hölder continuous functions "from below"

We assume that we have an $\alpha$-Hölder continuous function $f$ on the 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.