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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Extremal problems of Chebyshev type
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by Franz Peherstorfer PDF
Proc. Amer. Math. Soc. 137 (2009), 2351-2361 Request permission

Abstract:

Let $a \in \mathbb {C} \setminus [-1,1]$ be given. We consider the problem of finding $\sup |p(a)|$ among all polynomials $p$ with complex coefficients of degree less than or equal to $n$ with $\max _{-1\leq x \leq 1}|p(x)| \leq 1$. We derive an asymptotic expression for the extremal polynomial and for the extremal value in terms of elementary functions. The solution is based on the description of Zolotarev polynomials with respect to square root polynomial weights.
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Additional Information
  • Franz Peherstorfer
  • Affiliation: Abteilung für Dynamische Systeme und Approximationstheorie, Institut für Analysis, Johannes Kepler Universität Linz, Altenberger Strasse, 69, 4040 Linz, Austria
  • Email: franz.peherstorfer@jku.at
  • Received by editor(s): December 4, 2007
  • Received by editor(s) in revised form: September 18, 2008
  • Published electronically: January 13, 2009
  • Additional Notes: The author was supported by the Austrian Science Fund FWF, project no. P20413-N18
  • Communicated by: Peter A. Clarkson
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 2351-2361
  • MSC (2000): Primary 41A29; Secondary 33C45, 41A60
  • DOI: https://doi.org/10.1090/S0002-9939-09-09771-8
  • MathSciNet review: 2495269