On the sharpness of the weighted Bernstein-Walsh inequality, with applications to the superlinear convergence of conjugate gradients

Abstract: In this paper we show that the weighted Bernstein-Walsh inequality in logarithmic potential theory is sharp up to some new universal constant, provided that the external field is given by a logarithmic potential. Our main tool for such results is a new technique of discretization of logarithmic potentials, where we take the same starting point as in earlier work of Totik and of Levin & Lubinsky, but add an important new ingredient, namely some new mean value property for the cumulative distribution function of the underlying measure. As an application, we revisit the work of Beckermann & Kuijlaars on the superlinear convergence of conjugate gradients. These authors have determined the asymptotic convergence factor for sequences of systems of linear equations with an asymptotic eigenvalue distribution. There was some numerical evidence to let conjecture that the integral mean of Green functions occurring in their work should also allow to give inequalities for the rate of convergence if one makes a suitable link between measures and the eigenvalues of a single matrix of coefficients. We prove this conjecture , at least for a class of measures which is of particular interest for applications.