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Proof of a conjecture of Erdős about the longest polynomial

Author: B. D. Bojanov
Journal: Proc. Amer. Math. Soc. 84 (1982), 99-103
MSC: Primary 41A17
Addendum: Proc. Amer. Math. Soc. 89 (1983), 188.
MathSciNet review: 633287
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Abstract: In 1939 P. Erdös conjectured that the Chebyshev polynomial $ {T_n}(x)$ has a maximal arc-length in $ [ - 1,1]$ among the polynomials of degree $ n$ which are bounded by 1 in $ [ - 1,1]$. We prove this conjecture for every natural $ n$.

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

  • [1] B. D. Bojanov, An extension of the Markov inequality, J. Approx. Theory (submitted). MR 662166 (83h:41013)
  • [2] P. Erdös, An extremum-problem concerning trigonometric polynomials, Acta Sci. Math. Szeged 9 (1939), 113-115.
  • [3] T. J. Rivlin, The Chebyshev polynomials, Wiley, New York, 1974. MR 0450850 (56:9142)
  • [4] J. Szabados, On some extremum problems for polynomials, Proc. Conf. Approximation and Function Spaces (Gdansk 1979), PWN, Warsaw (to appear). MR 649472 (83h:42004)

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Keywords: Arc-length, Chebyshev polynomial, extremal problem for polynomials
Article copyright: © Copyright 1982 American Mathematical Society

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