  We consider quantum walks with position dependent coin on 1D lattice
$\mathbb{Z}$. The dispersive estimate $\|U^tP_c u_0\|_{l^\infty}\lesssim
(1+|t|)^{-1/3} \|u_0\|_{l^1}$ is shown under $l^{1,1}$ perturbation for the
generic case and $l^{1,2}$ perturbation for the exceptional case, where $U$ is
the evolution operator of a quantum walk and $P_c$ is the projection to the
continuous spectrum. This is an analogous result for Schr\"odinger operators
and discrete Schr\"odinger operators. The proof is based on the estimate of
oscillatory integrals expressed by Jost solutions.
