The failure of Bernstein's theorem for polynomials on C(K) spaces
type: Item
instance of: scholarly article
title: The failure of Bernstein's theorem for polynomials on C(K) spaces
zbMATH Open document ID: 0642.41006
DOI: 10.1016/0021-9045(87)90029-3
author: B. George
published in: Journal of Approximation Theory
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://bura.brunel.ac.uk/handle/2438/2281
review text: Bernstein's theorem states that if \(p: {\mathbb{C}}\to {\mathbb{C}}\) is a polynomial of degree m, then its derivative p' satisfies \(\| p'\|_{\infty}\leq m\| p\|_{\infty}\), where \(\| \|_{\infty}\) denotes the supremum norm taken over the unit disc. \textit{L. A. Harris}, Bounds on the derivatives of holomorphic functions of vectors, in C. R. Colloq. d'analyse, Rio de Janeiro, 1972, 145-163 (1975; Zbl 0315.46040) proved an analogous inequality for the Fréchet derivative of a polynomial on Hilbert space. He also asked whether there is a similar result for polynomials on C(K) spaces. The purpose of the present note is to give a negative answer, even for polynomials of degree two.
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 4046314
zbMATH Keywords: Bernstein's theorem
zbMATH Keywords: Fréchet derivative
zbMATH Keywords: Hilbert space
reviewed by: H. R. Dowson
MaRDI profile type: Publication
OpenAlex ID: W2095575007
cites work: Q3943468
cites work: Q4076354
cites work: Q3944558
cites work: On Bounds of Polynomials in Hyperspheres and Frechet-Michal Derivatives in Real and Complex Normed Linear Spaces
cites work: Estimates for polynomial norms on Lp(μ) spaces
cites work: Polarization and the two-dimensional Grothendieck inequality
cites work: On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory. (Appendix by S. Kaijser and N. Th. Varopoulos.)
description: scientific article; zbMATH DE number 4046314