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

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:155

Letter

Damelin, SB

关键词: logarithmic potential;    maximum modulus;    polynomial;   

DOI  :  10.1016/S0377-0427(02)00908-1

来源: Elsevier

【 摘 要 】

Let Sigma subset of or equal to C be a closed set of positive capacity at each point in Sigma and w : Sigma --> [0, infinity) a continuous, weight with \z\w(z) --> 0, \z\ --> infinity, z is an element of Sigma if Sigma 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 S-w denotes the support of the equilibrium measure for w, then parallel toP(n)w(n)parallel to(Sigma) = parallel toP(n)w(n)parallel tos(w) for any polynomial P-n with deg P-n less than or equal to n. This does not rule out the possibility that \P(n)w(n)\ may attain a maximum outside S-w. We prove that if in addition, Sigma is regular with respect to the Dirichlet problem on C and if it coincides with its outer boundary, then all points where \P(n)w(n)\ attain their maxima must lie in S-w. The case when Sigma subset of or equal to 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 Sigma cannot in general be relaxed. (C) 2002 Elsevier Science B.V. All rights reserved.

【 授权许可】

Free