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Remez-type inequality for discrete sets

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Abstract

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 Q n1 ⊂ ℝn can be bounded through the maximum of its absolute value on any subset Z ⊂ Q n1 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 ω d (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.

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  1. Department of Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    Y. Yomdin

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  1. Y. Yomdin
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Yomdin, Y. Remez-type inequality for discrete sets. Isr. J. Math. 186, 45–60 (2011). https://doi.org/10.1007/s11856-011-0131-4

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  • DOI: https://doi.org/10.1007/s11856-011-0131-4

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