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Extremal Polynomials and Sets of Minimal Capacity

Christiansen, Jacob Stordal LU ; Eichinger, Benjamin LU and Rubin, Olof LU orcid (2024) In Constructive Approximation
Abstract
This article examines the asymptotic behavior of the Widom factors, denoted Wn, for Chebyshev polynomials of finite unions of Jordan arcs. We prove that, in contrast to Widom’s proposal in Widom (Adv Math 3:127–232, 1969), when dealing with a single smooth Jordan arc, Wn converges to 2 exclusively when the arc is a straight line segment. Our main focus is on analysing polynomial preimages of the interval [-2,2], and we provide a complete description of the asymptotic behavior of Wn for symmetric star graphs and quadratic preimages of [-2,2]. We observe that in the case of star graphs, the Chebyshev polynomials and the polynomials orthogonal with respect to equilibrium measure share the same norm asymptotics, suggesting a potential... (More)
This article examines the asymptotic behavior of the Widom factors, denoted Wn, for Chebyshev polynomials of finite unions of Jordan arcs. We prove that, in contrast to Widom’s proposal in Widom (Adv Math 3:127–232, 1969), when dealing with a single smooth Jordan arc, Wn converges to 2 exclusively when the arc is a straight line segment. Our main focus is on analysing polynomial preimages of the interval [-2,2], and we provide a complete description of the asymptotic behavior of Wn for symmetric star graphs and quadratic preimages of [-2,2]. We observe that in the case of star graphs, the Chebyshev polynomials and the polynomials orthogonal with respect to equilibrium measure share the same norm asymptotics, suggesting a potential extension of the conjecture posed in Christiansen et al. (Oper Theory Adv Appl 289:301–319, 2022). Lastly, we propose a possible connection between the S-property and Widom factors converging to 2. (Less)
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publication status
published
subject
in
Constructive Approximation
publisher
Springer
external identifiers
  • scopus:85203528056
ISSN
1432-0940
DOI
10.1007/s00365-024-09690-4
project
Chebyshev polynomials - Complexities in the complex plane
language
English
LU publication?
yes
id
90fa7a8f-f296-48ed-8226-5a7372ebd410
date added to LUP
2024-09-11 18:19:12
date last changed
2025-04-04 13:57:07
@article{90fa7a8f-f296-48ed-8226-5a7372ebd410,
  abstract     = {{This article examines the asymptotic behavior of the Widom factors, denoted Wn, for Chebyshev polynomials of finite unions of Jordan arcs. We prove that, in contrast to Widom’s proposal in Widom (Adv Math 3:127–232, 1969), when dealing with a single smooth Jordan arc, Wn converges to 2 exclusively when the arc is a straight line segment. Our main focus is on analysing polynomial preimages of the interval [-2,2], and we provide a complete description of the asymptotic behavior of Wn for symmetric star graphs and quadratic preimages of [-2,2]. We observe that in the case of star graphs, the Chebyshev polynomials and the polynomials orthogonal with respect to equilibrium measure share the same norm asymptotics, suggesting a potential extension of the conjecture posed in Christiansen et al. (Oper Theory Adv Appl 289:301–319, 2022). Lastly, we propose a possible connection between the S-property and Widom factors converging to 2.}},
  author       = {{Christiansen, Jacob Stordal and Eichinger, Benjamin and Rubin, Olof}},
  issn         = {{1432-0940}},
  language     = {{eng}},
  month        = {{09}},
  publisher    = {{Springer}},
  series       = {{Constructive Approximation}},
  title        = {{Extremal Polynomials and Sets of Minimal Capacity}},
  url          = {{http://dx.doi.org/10.1007/s00365-024-09690-4}},
  doi          = {{10.1007/s00365-024-09690-4}},
  year         = {{2024}},
}