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

Polynomial of best uniform approximation to $x^{-1}$ and smoothing in two-level methods