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

Applications of Fourier analysis in homogenization of Dirichlet problem I. Pointwise Estimates