Let $\alpha\in(0,1)$ and $\eta\in\Lambda_0^\alpha(\mathbb{R})$ be a compactly supported Hölder continuous function of order $\alpha$. I would like to show that, for any $n\in\mathbb{N}$, it is possible to decompose $$\eta=f+g$$ in such a way that $f\in C^n(\mathbb{R})$ and $||f||_{C^n}=O(R^C)$, and $g\in L^\infty(\mathbb{R})$ with $\|g\|_{L^\infty}=O(R^{-1})$.

Here $C$ is a universal constant. On the other hand, the real parameter $R$ can be chosen as large as we want (at the expense of increasing $\|f\|_{C^n}$).

Thank you!