A proof of Markov's theorem for polynomials on Banach spaces

Lawrence A. Harris

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


Our object is to present an independent proof of the extension of V.A. Markov's theorem to Gâteaux derivatives of arbitrary order for continuous polynomials on any real normed linear space. The statement of this theorem differs little from the classical case for the real line except that absolute values are replaced by norms. Our proof depends only on elementary computations and explicit formulas and gives a new proof of the classical theorem as a special case. Our approach makes no use of the classical polynomial inequalities usually associated with Markov's theorem. Instead, the essential ingredients are a Lagrange interpolation formula for the Chebyshev nodes and a Christoffel-Darboux identity for the corresponding bivariate Lagrange polynomials. We use these tools to extend a single variable inequality of Rogosinski to the case of two real variables. The general Markov theorem is an easy consequence of this.

Original languageEnglish
Pages (from-to)374-381
Number of pages8
JournalJournal of Mathematical Analysis and Applications
Issue number1
StatePublished - Aug 2010


  • Bivariate interpolation
  • Chebyshev nodes
  • Christoffel-Darboux identity
  • Normed linear spaces
  • Polynomial operators

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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