On the maximum modulus of weighted polynomials in the plane, a theorem of Rakhmanov, Mhaskar and Saff revisited

摘要

Let Σ⊆C be a closed set of positive capacity at each point in Σ and w:Σ→[0,∞) a continuous, weight with |z|w(z)→0,|z|→∞,z∈Σ if Σ is unbounded. Assume further that the set where w is positive is of positive capacity. A classical theorem, obtained independently by Rakhmanov and Mhaskar and Saff says that if Sw denotes the support of the equilibrium measure for w, then ||Pnwn||Σ=||Pnwn||Sw for any polynomial Pn with degPn⩽n. This does not rule out the possibility that |Pnwn| may attain a maximum outside Sw. We prove that if in addition, Σ is regular with respect to the Dirichlet problem on C and if it coincides with its outer boundary, then all points where |Pnwn| attain their maxima must lie in Sw. The case when Σ⊆R consists of a finite union of finite or infinite intervals is due to Lorentz, von Golitschek and Makovoz. Counter examples are given to show that our requirements on Σ cannot in general be relaxed.