  Algebraic multigrid (AMG) coarse spaces are commonly constructed so that they
exhibit the so-called weak approximation (WAP) property which is necessary and
sufficient condition for uniform two-grid convergence. This paper studies a
modification of such coarse spaces so that the modified ones provide
approximation in energy norm. Our modification is based on the projection in
energy norm onto an orthogonal complement of original coarse space. This
generally leads to dense modified coarse space matrices which is hence
computationally infeasible. To remedy this, based on the fact that the
projection involves inverse of a well-conditioned matrix, we use polynomials to
approximate the projection and, therefore, obtain a practical, sparse modified
coarse matrix and prove that the modified coarse space maintains
computationally feasible approximation in energy norm. We present some
numerical results for both, PDE discretization matrices as well as graph
Laplacian ones, which are in accordance with our theoretical results.
