Let $0<s<1$. I'm interested to find a class of smooth functions such that they are dense in $B_{p\infty}^{s}\left(\mathbb{R}\right)$ with $1 \leq p <+\infty$ with respect the norm $$ \left\Vert f\right\Vert _{B_{p\infty}^{s}\left(\mathbb{R}\right)}=\left\Vert f\right\Vert _{L^{p}\left(\mathbb{R}\right)}+\sup_{t>0}\frac{\omega_{p}\left(t;f\right)}{t^{s}} $$ where $\omega_{p}\left(t;f\right)$ is the modulus of continuty in $L^{p}$, and a (not necesseraly equal) class of smooth function dense in $B_{\infty \infty}^{s}\left(\mathbb{R}\right)$ with respect the norm $$ \left\Vert f\right\Vert _{B_{\infty\infty}^{s}\left(\mathbb{R}\right)}=\left\Vert f\right\Vert _{L^{\infty}\left(\mathbb{R}\right)}+\sup_{t>0}\frac{\omega_{\infty}\left(t;f\right)}{t^{s}}. $$ I know that the Schwartz functions are dense in $B_{pq}^{s}\left(\mathbb{R}\right)$ if $\max(p,q)<+\infty$ but this is not true if $p=+\infty$ or $q=+\infty$. For what concerning $B_{\infty q}^{s}\left(\mathbb{R}\right)$ I found that the smooth functions $C^{\infty}\left(\mathbb{R}\right)$ are dense but for the other cases I found nothing.
ADDENDUM: In a previous version of this question, I had conjectured that $C_{0}^{\infty}\left(\mathbb{R}\right)$ functions were dense in $B_{p\infty}^{s}\left(\mathbb{R}\right)$ and $B_{\infty \infty}^{s}\left(\mathbb{R}\right)$ but, as also pointed out by Marco, this is false.
