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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extremal problems of Chebyshev type


Author: Franz Peherstorfer
Journal: Proc. Amer. Math. Soc. 137 (2009), 2351-2361
MSC (2000): Primary 41A29; Secondary 33C45, 41A60
Published electronically: January 13, 2009
MathSciNet review: 2495269
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Abstract: Let $ a \in \mathbb{C} \setminus [-1,1]$ be given. We consider the problem of finding $ \sup \vert p(a)\vert$ among all polynomials $ p$ with complex coefficients of degree less than or equal to $ n$ with $ \max_{-1\leq x \leq 1}\vert p(x)\vert \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

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09771-8
PII: S 0002-9939(09)09771-8
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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.