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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 | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. N. I. Akhiezer, Lectures on Approximation Theory, 2nd ed., Akademie-Verlag, Berlin, 1967.
  • 2. R. Freund and S. Ruscheweyh, On a class of Chebyshev approximation problems which arise in connection with a conjugate gradient type method, Numer. Math. 48 (1986), 525-542. MR 839615 (87f:41048)
  • 3. S. Karlin and W. Studden, Tchebycheff Systems, Interscience Publ., Wiley and Sons, New York, 1966.MR 0204922 (34:4757)
  • 4. A. N. Kolmogorov and A. P. Yuskhevich (eds.), Mathematics of the 19th Century, Constructive Function Theory, Ordinary Differential Equations, Calculus of Variations, Theory of Finite Differences, Birkhäuser, Basel, 1998. MR 1634232 (99d:01022)
  • 5. M. Kreın and A. Nudel$ ^\prime$man, The Markov moment problem and extremal problems, Trans. of Math. Monographs 50, Amer. Math. Soc., Providence, Rhode Island, 1977. MR 0458081 (56:16284)
  • 6. F. Peherstorfer, Asymptotic representation of Zolotarev polynomials, J. London Math. Soc. (2) 74 (2006), 143-153. MR 2254557 (2008i:30040)
  • 7. P. Yuditskii, A complex extremal problem of Chebyshev type, J. Anal. Math. 77 (1999), 207-235. MR 1753486 (2001b:30038)

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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

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.

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