  In this paper we prove convergence results for the homogenization of the
Dirichlet problem with rapidly oscillating boundary data in convex polygonal
domains. Our analysis is based on integral representation of solutions. Under a
certain Diophantine condition on the boundary of the domain and smooth
coefficients we prove pointwise, as well as $L^p$ convergence results. For
larger exponents $p$ we prove that the $L^p$ convergence rate is close to
optimal. We shall also suggest several directions of possible generalization of
the result in this paper.
