Asymptotics of Polynomial Interpolation and the Bernstein Constants

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Abstract

It is well known that the interpolation error for | x| α, α> 0 in L [- 1 , 1] by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind can be represented in its limiting form by entire functions of exponential type. In this paper, we establish new asymptotic bounds for these quantities when α tends to infinity. Moreover, we present some explicit constructions for near best approximation polynomials to | x| α, α> 0 in the L norm which are based on the Chebyshev interpolation process. The resulting formulas possibly indicate a general approach towards the structure of the associated Bernstein constants.

Original languageEnglish
Article number100
Pages (from-to)76-100
Number of pages25
JournalResults in Mathematics
Volume76
Issue number2
DOIs
Publication statusPublished - 19 Apr 2021

Fields of Science and Technology Classification 2012

  • 101 Mathematics

Keywords

  • Bernstein constants
  • best uniform approximation
  • Chebyshev nodes
  • entire functions of exponential type
  • higher order asymptotics
  • Watson-lemma

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