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In this paper, a uniformly elliptic second order boundary value problem in two dimensions is discretized by the p-version of the finite element method. An inexact Dirichlet--Dirichlet domain decomposition preconditioner for the system of linear algebraic equations is investigated. The ingredients of such a preconditioner are a preconditioner for the Schur-complement, a preconditioner for the subdomains, and an extension operator operating from the edges of the elements into their interior. Using methods of multiresolution analysis, we propose a new method to compute the extension efficiently. We prove that this type of extension is optimal, i.e., the $H^1(\Omega)$-norm of the extended function is bounded by the $H^{0.5}(\partial\Omega)$-norm of the given function. Numerical experiments show the optimal performance of the described extension.