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Asymptotics of Chebyshev polynomials, I: subsets of ℝ

Christiansen, Jacob S. and Simon, Barry and Zinchenko, Maxim (2017) Asymptotics of Chebyshev polynomials, I: subsets of ℝ. Inventiones Mathematicae, 208 (1). pp. 217-245. ISSN 0020-9910. doi:10.1007/s00222-016-0689-x.

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We consider Chebyshev polynomials, T_n(z), for infinite, compact sets e⊂ℝ (that is, the monic polynomials minimizing the sup-norm, ||T_n||_e, on e). We resolve a 45+ year old conjecture of Widom that for finite gap subsets of R, his conjectured asymptotics (which we call Szegő–Widom asymptotics) holds. We also prove the first upper bounds of the form ||T_n||_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⊂ℝ that obey a Parreau–Widom condition.

Item Type:Article
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URLURL TypeDescription Paper ReadCube access
Simon, Barry0000-0003-2561-8539
Additional Information:© 2016 Springer-Verlag Berlin Heidelberg. Received: 18 October 2015; Accepted: 30 April 2016; First Online: 19 September 2016. B. Simon’s research was supported in part by NSF Grant DMS-1265592 and in part by Israeli BSF Grant No. 2010348. M. Zinchenko’s research was supported in part by Simons Foundation Grant CGM-281971.
Funding AgencyGrant Number
Binational Science Foundation (USA-Israel)2010348
Simons FoundationCGM-281971
Issue or Number:1
Classification Code:Mathematics Subject Classification: 41A50; 30E15; 30C10
Record Number:CaltechAUTHORS:20161109-142515049
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Official Citation:Christiansen, J.S., Simon, B. & Zinchenko, M. Invent. math. (2017) 208: 217. doi:10.1007/s00222-016-0689-x
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:71891
Deposited By: Tony Diaz
Deposited On:09 Nov 2016 22:38
Last Modified:11 Nov 2021 04:52

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