  In this paper, we focus on zero-filter limit problem for the Camassa-Holm
equation in the more general Besov spaces. We prove that the solution of the
Camassa-Holm equation converges strongly in $L^\infty(0,T;B^s_{2,r}(\R))$ to
the inviscid Burgers equation as the filter parameter $\alpha$ tends to zero
with the given initial data $u_0\in B^s_{2,r}(\R)$. Moreover, we also show that
the zero-filter limit for the Camassa-Holm equation does not converges
uniformly with respect to the initial data in $B^s_{2,r}(\R)$.
