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Szegő–Widom asymptotics of Chebyshev polynomials on circular arcs

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Szegő–Widom asymptotics of Chebyshev polynomials on circular arcs. / Eichinger, Benjamin.
In: Journal of Approximation Theory, Vol. 217, 01.05.2017, p. 15-25.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Eichinger, B 2017, 'Szegő–Widom asymptotics of Chebyshev polynomials on circular arcs', Journal of Approximation Theory, vol. 217, pp. 15-25. https://doi.org/10.1016/j.jat.2017.02.005

APA

Eichinger, B. (2017). Szegő–Widom asymptotics of Chebyshev polynomials on circular arcs. Journal of Approximation Theory, 217, 15-25. https://doi.org/10.1016/j.jat.2017.02.005

Vancouver

Eichinger B. Szegő–Widom asymptotics of Chebyshev polynomials on circular arcs. Journal of Approximation Theory. 2017 May 1;217:15-25. Epub 2017 Mar 19. doi: 10.1016/j.jat.2017.02.005

Author

Eichinger, Benjamin. / Szegő–Widom asymptotics of Chebyshev polynomials on circular arcs. In: Journal of Approximation Theory. 2017 ; Vol. 217. pp. 15-25.

Bibtex

@article{e53d468414ad47329f1510e810e0d6b4,
title = "Szeg{\H o}–Widom asymptotics of Chebyshev polynomials on circular arcs",
abstract = "Thiran and Detaille give an explicit formula for the asymptotics of the sup-norm of the Chebyshev polynomials on a circular arc. We give the so-called Szeg{\H o}–Widom asymptotics for this domain, i.e., explicit expressions for the asymptotics of the corresponding extremal polynomials. Moreover, we solve a similar problem with respect to the upper envelope of a family of polynomials uniformly bounded on this arc. That is, we give explicit formulas for the asymptotics of the error of approximation as well as of the extremal functions. Our computations show that in the proper normalization the limit of the upper envelope represents the diagonal of a reproducing kernel of a certain Hilbert space of analytic functions. Due to Garabedian, the analytic capacity in an arbitrary domain is the diagonal of the corresponding Szeg{\H o} kernel. We do not know any result of this kind with respect to upper envelopes of polynomials. If this is a general fact or a specific property of the given domain, we rise as an open question.",
keywords = "Chebyshev polynomials, Reproducing kernels, Szeg{\H o}–Widom asymptotics, Uniform approximation",
author = "Benjamin Eichinger",
year = "2017",
month = may,
day = "1",
doi = "10.1016/j.jat.2017.02.005",
language = "English",
volume = "217",
pages = "15--25",
journal = "Journal of Approximation Theory",
issn = "0021-9045",
publisher = "Academic Press Inc.",

}

RIS

TY - JOUR

T1 - Szegő–Widom asymptotics of Chebyshev polynomials on circular arcs

AU - Eichinger, Benjamin

PY - 2017/5/1

Y1 - 2017/5/1

N2 - Thiran and Detaille give an explicit formula for the asymptotics of the sup-norm of the Chebyshev polynomials on a circular arc. We give the so-called Szegő–Widom asymptotics for this domain, i.e., explicit expressions for the asymptotics of the corresponding extremal polynomials. Moreover, we solve a similar problem with respect to the upper envelope of a family of polynomials uniformly bounded on this arc. That is, we give explicit formulas for the asymptotics of the error of approximation as well as of the extremal functions. Our computations show that in the proper normalization the limit of the upper envelope represents the diagonal of a reproducing kernel of a certain Hilbert space of analytic functions. Due to Garabedian, the analytic capacity in an arbitrary domain is the diagonal of the corresponding Szegő kernel. We do not know any result of this kind with respect to upper envelopes of polynomials. If this is a general fact or a specific property of the given domain, we rise as an open question.

AB - Thiran and Detaille give an explicit formula for the asymptotics of the sup-norm of the Chebyshev polynomials on a circular arc. We give the so-called Szegő–Widom asymptotics for this domain, i.e., explicit expressions for the asymptotics of the corresponding extremal polynomials. Moreover, we solve a similar problem with respect to the upper envelope of a family of polynomials uniformly bounded on this arc. That is, we give explicit formulas for the asymptotics of the error of approximation as well as of the extremal functions. Our computations show that in the proper normalization the limit of the upper envelope represents the diagonal of a reproducing kernel of a certain Hilbert space of analytic functions. Due to Garabedian, the analytic capacity in an arbitrary domain is the diagonal of the corresponding Szegő kernel. We do not know any result of this kind with respect to upper envelopes of polynomials. If this is a general fact or a specific property of the given domain, we rise as an open question.

KW - Chebyshev polynomials

KW - Reproducing kernels

KW - Szegő–Widom asymptotics

KW - Uniform approximation

U2 - 10.1016/j.jat.2017.02.005

DO - 10.1016/j.jat.2017.02.005

M3 - Journal article

AN - SCOPUS:85015705613

VL - 217

SP - 15

EP - 25

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

SN - 0021-9045

ER -