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Asymptotics of Chebyshev polynomials, I : subsets of R

Christiansen, Jacob S. LU ; Simon, Barry and Zinchenko, Maxim (2017) In Inventiones Mathematicae 208(1). p.217-245
Abstract

We consider Chebyshev polynomials, (Formula presented.), for infinite, compact sets (Formula presented.) (that is, the monic polynomials minimizing the (Formula presented.)-norm, (Formula presented.), on (Formula presented.)). We resolve a (Formula presented.) year old conjecture of Widom that for finite gap subsets of (Formula presented.), his conjectured asymptotics (which we call Szegő–Widom asymptotics) holds. We also prove the first upper bounds of the form (Formula presented.) (where (Formula presented.) is the logarithmic capacity of (Formula presented.)) for a class of (Formula presented.)’s with an infinite number of components, explicitly for those (Formula presented.) that obey a Parreau–Widom condition.

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author
; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Inventiones Mathematicae
volume
208
issue
1
pages
29 pages
publisher
Springer
external identifiers
  • scopus:84988369421
  • wos:000398170000004
ISSN
0020-9910
DOI
10.1007/s00222-016-0689-x
language
English
LU publication?
yes
id
68bf08f1-a9da-4d03-bd49-7f218b0f65af
date added to LUP
2016-11-03 08:58:05
date last changed
2025-01-12 14:16:12
@article{68bf08f1-a9da-4d03-bd49-7f218b0f65af,
  abstract     = {{<p>We consider Chebyshev polynomials, (Formula presented.), for infinite, compact sets (Formula presented.) (that is, the monic polynomials minimizing the (Formula presented.)-norm, (Formula presented.), on (Formula presented.)). We resolve a (Formula presented.) year old conjecture of Widom that for finite gap subsets of (Formula presented.), his conjectured asymptotics (which we call Szegő–Widom asymptotics) holds. We also prove the first upper bounds of the form (Formula presented.) (where (Formula presented.) is the logarithmic capacity of (Formula presented.)) for a class of (Formula presented.)’s with an infinite number of components, explicitly for those (Formula presented.) that obey a Parreau–Widom condition.</p>}},
  author       = {{Christiansen, Jacob S. and Simon, Barry and Zinchenko, Maxim}},
  issn         = {{0020-9910}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{217--245}},
  publisher    = {{Springer}},
  series       = {{Inventiones Mathematicae}},
  title        = {{Asymptotics of Chebyshev polynomials, I : subsets of R}},
  url          = {{http://dx.doi.org/10.1007/s00222-016-0689-x}},
  doi          = {{10.1007/s00222-016-0689-x}},
  volume       = {{208}},
  year         = {{2017}},
}