  In this paper, we provide a proof that functions belonging to Besov spaces
$B^{r}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d)$, $q\in [1,\infty)$, $r\in(0,1)$,
satisfy the following formula under a certain condition:
  \begin{equation} \label{eq:main result in abstract} \lim_{{\epsilon}\to
0^+}\frac{1}{|\ln{\epsilon}|}\left[u_{\epsilon}\right]^q_{W^{r,q}(\mathbb{R}^N,\mathbb{R}^d)}=N\lim_{{\epsilon}\to
0^+}\int_{\mathbb{R}^N}\frac{1}{{\epsilon}^N}\int_{B_{\epsilon}(x)}\frac{|u(x)-u(y)|^q}{|x-y|^{rq}}dydx.
\end{equation} Here, $\left[\cdot\right]_{W^{r,q}}$ represents the Gagliardo
seminorm, and $u_{\epsilon}$ denotes the convolution of $u$ with a mollifier
$\eta_{(\epsilon)}(x):=\frac{1}{\epsilon^N}\eta\left(\frac{x}{\epsilon}\right)$,
$\eta\in W^{1,1}(\mathbb{R}^N),\int_{\mathbb{R}^N}\eta(z)dz=1$. Furthermore, we
prove that every function $u$ in $BV(\mathbb{R}^N,\mathbb{R}^d)\cap
B^{1/p}_{p,\infty}(\mathbb{R}^N,\mathbb{R}^d),p\in(1,\infty),$ satisfies
\begin{multline} \lim_{\epsilon\to
0^+}\frac{1}{|\ln{\epsilon}|}\left[u_{\epsilon}\right]^q_{W^{1/q,q}(\mathbb{R}^N,\mathbb{R}^d)}=N\lim_{{\epsilon}\to
0^+}\int_{\mathbb{R}^N}\frac{1}{{\epsilon}^N}\int_{B_{\epsilon}(x)}\frac{|u(x)-u(y)|^q}{|x-y|}dydx
=\left(\int_{S^{N-1}}|z_1|~d\mathcal{H}^{N-1}(z)\right)\int_{\mathcal{J}_u}
\Big|u^+(x)-u^-(x)\Big|^q d\mathcal{H}^{N-1}(x), \end{multline} for every
$1<q<p$. Here $u^+,u^-$ are the one-sided approximate limits of $u$ along the
jump set $\mathcal{J}_u$.
