An upper bound for the norm of the Chebyshev polynomial on two intervals

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Abstract

Let E:=[−1,α]∪[β,1], −1<α<β<1, be the union of two real intervals and consider the Chebyshev polynomial of degree n on E, that is, that monic polynomial which is minimal with respect to the supremum norm on E. For its norm, called the n-th Chebyshev number of E, an upper bound in terms of elementary functions of α and β is given. The proof is based on results of N.I. Achieser in the 1930s in which the norm is estimated with the help of Zolotarev's transformation using Jacobi's elliptic and theta functions.

Original languageEnglish
Pages (from-to)871-883
Number of pages13
JournalJournal of Mathematical Analysis and Applications
Volume445
Issue number1
DOIs
Publication statusPublished - 1 Jan 2017

Keywords

  • Chebyshev number
  • Chebyshev polynomial
  • Jacobi's elliptic functions
  • Jacobi's theta functions
  • Logarithmic capacity
  • Two intervals

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