2009/11/10 by Yosef Yomdin, Yomdin, Y. · 1 citation
Mathematics · #Mathematical functions and polynomials #Functional Equations Stability Results #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.0911.1937
The classical Remez inequality bounds the maximum of the absolute value of a polynomial P(x) of degree d on [-1,1] through the maximum of its absolute value on any subset Z of positive measure in [-1,1]. Similarly, in several variables the maximum of the absolute value of a polynomial P(x) of degree d on the unit cube Qn1 ⊂ \mathbb Rn can be bounded through the maximum of its absolute value on any subset Z⊂ Qn1 of positive n-measure. The main result of this paper is that the n-measure in the Remez inequality can be replaced by a certain geometric invariant ωd(Z) which can be effectively estimated in terms of the metric entropy of Z and which may be nonzero for discrete and even finite sets Z.