Approximation of $C^\alpha(T^2)$ in $L^1$ by a $C^k$ function

Question

Consider a Holder function $f \in C^\alpha(\mathbb{T}^2, \mathbb{R})$, $\alpha \in (0,1)$. I would like to approximate $f$ with $f_\epsilon \in C^k(\mathbb{T}^2, \mathbb{R})$, $k \in \mathbb{N}$, in the $L^1(\mathbb{T}^2)$ norm.

I was wondering how one does such approximation? In particular, how one get estimates on $\|f-f_\epsilon\|_{L^1}$ and $\|f_\epsilon\|_{C^k}$ in terms of $\epsilon$?

Thank you in advance.