Consider either a Besov space $B_{p,q}^s$ or a Triebel-Lizorkin space $F_{p,q}^s$ with parameters $p,q \in [1,\infty)$ and $s \in \mathbb{R}$. It is known that the Schwartz space $\mathcal{S}(\mathbb{R})$ is dense in $A_{p,q}^s$ for $A \in \{B,F\}$. I was looking for a "concrete" approximation. From my perspective, the most natural idea is to use convolutions: Fix $f \in A_{p,q}^s$ with $A \in \{B,F\}$. For $\chi \in C_c^{\infty}(\mathbb{R}^n)$ with $\chi \geq 0$ and $\int \chi(x) \, dx =1$, define $\chi_k(x) := k^d \chi(kx)$. Then
$$f_k := f* \chi_k$$
should converge to $f$ in $A_{p,q}^s$, right? Moreover, $f_k$ is a Schwartz function, and we are done. Does anybody know a reference for this result? I checked the books by Triebel and also the book by Sawano, but I couldn't find it (probably because it is "obvious").
