For any sequence $x=(x_k)^\infty_{k=1}\in \omega$ and any subset $X$ of $\omega$, we write $\Delta x=(\Delta x_k)^\infty_{k=1}=(x_k-x_{k+1})^\infty_{k=1}$ and $\Delta X=\{x\in\omega:\Delta x\in X\}$. Let $p=(p_k)^\infty_{k=1}$ and $q=(q_k)^\infty_{k=1}$ be bounded sequences of positive reals. We determine the $\beta$-duals of the sets $\Delta c_0(p)$, $\Delta c(p)$ and $\Delta\ell_\infty(p)$. Furthermore, we characterize the matrix classes $(\Delta X,Y)$ and $(\Delta X,\Delta Y)$ for $X=c_0(p),c(p),l_\infty(p)$ and $Y=c_0(q),c(q),l_\infty(q)$.