We apply multiresolution techniques to study the effective properties of boundary value problems. Conditions under which boundary values are preserved in the effective ('homogenized') formulation are developed and discussed. Relations between the eigenfunctions and eigenvalues of the generic formulation and those of the effective formulation are explored and their dependence on the micro-scale is studied. Applications to the construction of effective Green functions in a complex lamination are discussed. The analytic results are demonstrated via numerical computations.