  We derive a three-term recurrence relation for computing the polynomial of
best approximation in the uniform norm to $x^{-1}$ on a finite interval with
positive endpoints. As application, we consider two-level methods for scalar
elliptic partial differential equation (PDE), where the relaxation on the fine
grid uses the aforementioned polynomial of best approximation. Based on a new
smoothing property of this polynomial smoother that we prove, combined with a
proper choice of the coarse space, we obtain as a corollary, that the
convergence rate of the resulting two-level method is uniform with respect to
the mesh parameters, coarsening ratio and PDE coefficient variation.
