Bounds for polynomials with a unit discrete norm

• 成果类型: Article
• 署名作者: Rakhmanov, E. A.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2007.165.55
• 发表日期: 2007
• 页码: 55-88
• 关键词: orthogonal polynomials; extremal polynomials; sets

摘要

Let E be the set of N equidistant points in (- 1, 1) and P-n, (E) be the set of all polynomials P of degree <= n with max {vertical bar P(zeta)vertical bar,zeta is an element of E} <= 1. We prove that [GRAPHICS] T where n < N and C is an absolute constant. The result is essentially sharp. Bounds for K-n,K-N(z), z E C, uniform for n < N, are also obtained. The method of proof of those results is a general one. It allows one to obtain sharp, or sharp up to a log N factor, bounds for K-n,K-N under rather general assumptions on E (#E = N). A model result is announced for a class of sets E. Main components of the method are discussed in some detail in the process of investigating the case of equally spaced points.