Abstract

We provide sharp Bernstein and Markov inequalities for 2-homogeneous polynomials on the square □⊂R2\Box\subset {\mathbb R}^2 with vertices (0,0)(0,0), (1,0)(1,0), (1,1)(1,1) and (0,1)(0,1). If P(2□){\mathcal P}(^2\Box) is the space of such polynomials, we also find the polarization constant of P(2□){\mathcal P}(^2\Box) and the unconditional constant for the canonical basis of P(2□){\mathcal P}(^2\Box). All the results are obtained by means of the Krein-Milman Theorem, using a characterization of the extreme 2-homogeneous polynomials on □\Box which is also given in the paper.

Author information

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José Luis Gámez-Merino

Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain

jlgamez@mat.ucm.es

Gustavo A. Muñoz-Fernández

Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain

gustavo_fernandez@mat.ucm.es

Viktor M. Sánchez

Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain

victorms@mat.ucm.es

Juan B. Seoane-Sepúlveda

Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain

jseoane@mat.ucm.es

Keywords

  • Convexity
  • extreme points
  • polynomial norms
  • Bernstein and Markov inequalities
  • polarization constants

Mathematics Subject Classification

41A17; 26D05, 52A21, 46B04

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Suggested citation

J. L. Gámez-Merino, G. A. Muñoz-Fernández, V. M. Sánchez, J. B. Seoane-Sepúlveda. “Inequalities for Polynomials on the Unit Square via the Krein-Milman Theorem.” Journal of Convex Analysis 20 (2013), No. 1, 125–142. https://doi.org/10.68381/jca20008

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.