  We construct a distorted Fourier transformation associated with the
multi-dimensional quantum walk. In order to avoid the complication of
notations, almost all of our arguments are restricted to two dimensional
quantum walks (2DQWs) without loss of generality. The distorted Fourier
transformation characterizes generalized eigenfunctions of the time evolution
operator of the QW. The 2DQW which will be considered in this paper has an
anisotropy due to the definition of the shift operator for the free QW. Then we
define an anisotropic Banach space as a modified Agmon-H\"{o}rmander's
$\mathcal{B}^*$ space and we derive the asymptotic behavior at infinity of
generalized eigenfunctions in these spaces. The scattering matrix appears in
the asymptotic expansion of generalized eigenfunctions.
