A bivariate Markov inequality for Chebyshev polynomials of the second kind

Lawrence A. Harris

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1 Scopus citations

Abstract

This note presents a Markov-type inequality for polynomials in two variables where the Chebyshev polynomials of the second kind in either one of the variables are extremal. We assume a bound on a polynomial at the set of even or odd Chebyshev nodes with the boundary nodes omitted and obtain bounds on its even or odd order directional derivatives in a critical direction. Previously, the author has given a corresponding inequality for Chebyshev polynomials of the first kind and has obtained the extension of V.A. Markov's theorem to real normed linear spaces as an easy corollary.To prove our inequality we construct Lagrange polynomials for the new class of nodes we consider and give a corresponding Christoffel-Darboux formula. It is enough to determine the sign of the directional derivatives of the Lagrange polynomials.

Original languageEnglish
Pages (from-to)1806-1814
Number of pages9
JournalJournal of Approximation Theory
Volume163
Issue number12
DOIs
StatePublished - Dec 2011

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • General Mathematics
  • Applied Mathematics

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