Asymptotics of Chebyshev polynomials, I: subsets of R

• 成果类型: Article
• 署名作者: Christiansen, Jacob S.; Simon, Barry; Zinchenko, Maxim
• 署名单位: Lund University; California Institute of Technology; California Institute of Technology; University of New Mexico
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0689-x
• 发表日期: 2017
• 页码: 217-245
• 关键词: extremal polynomials; algebraic equations; inverse images; UNIVERSALITY; intervals; spectrum; roots

摘要

We consider Chebyshev polynomials, T-n(z), for infinite, compact sets e subset of R (that is, the monic polynomials minimizing the sup-norm,parallel to T-n parallel to(e), on ). We resolve a 45+ year old conjecture of Widom that for finite gap subsets of R , his conjectured asymptotics (which we call Szego-Widom asymptotics) holds. We also prove the first upper bounds of the form parallel to T-n parallel to(e) <= QC(e)(n) (where C (e) is the logarithmic capacity of e) for a class of e's with an infinite number of components, explicitly for those e subset of R that obey a Parreau-Widom condition.