A proof of Markov's theorem for polynomials on Banach spaces
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- 作者：Lawrence A. Harris
- 关键词：Normed linear spaces ; Polynomial operators ; Bivariate interpolation ; Chebyshev nodes ; Christoffel– ; Darboux identity
- 刊名：Journal of Mathematical Analysis and Applications
- 出版年：2010
- 出版时间：1 August 2010
- 年：2010
- 卷：368
- 期：1
- 页码：374-381
- 全文大小：177 K
文摘
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.