  In a seminal work of A. N. Varchenko, the behavior at infinity of oscillatory
integrals with real analytic phase is precisely investigated by using the
theory of toric varieties based on the geometry of the Newton polyhedron of the
phase. The purpose of this paper is to generalize his results to the case that
the phase is contained in a certain class of $C^{\infty}$ functions. The key in
our analysis is a toric resolution of singularities in the above class of
$C^{\infty}$ functions. The properties of poles of local zeta functions, which
are closely related to the behavior of oscillatory integrals, are also studied
under the associated situation.
