In this paper, a uniformly elliptic second order boundary value problem in 2D is discretized by the p-version of the finite element method. An inexact Dirichlet-Dirichlet domain decomposition pre-conditioner for the system of linear algebraic equations is investigated. The ingredients of such a pre-conditioner are an pre-conditioner for the Schur complement, an preconditioner for the sub-domains and an extension operator operating from the edges of the elements into their interior. Using methods of multi-resolution analysis, we propose a new method in order to compute the extension efficiently. We prove that this type of extension is optimal, i.e. the H(Ω)-norm of the extended function is bounded by the H(∂Ω)-norm of the given function. Numerical experiments show the optimal performance of the described extension.