  In this paper we prove convergence results for homogenization problem for
solutions of partial differential system with rapidly oscillating Dirichlet
data. Our method is based on analysis of oscillatory integrals. In the
uniformly convex and smooth domain, and smooth operator and boundary data, we
prove pointwise convergence results, namely $$|u_{\e}(x)-u_0 (x)| \leq
C_{\kappa} \e^{(d-1)/2}\frac{1}{d(x)^{\kappa}}, \ \forall x\in D, \ \forall \
\kappa>d-1,$$ where $u_{\e}$ and $u_0$ are solutions of respectively
oscillating and homogenized Dirichlet problems, and $d(x)$ is the distance of
$x$ from the boundary of $D$. As a corollary for all $1\leq p <\infty$ we
obtain $L^p$ convergence rate as well.
