Remez-type inequality for discrete sets

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 n 1 ⊂ R n can be bounded through the maximum of its absolute value on any subset Z ⊂ Q n 1 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.

Norming Sets and Related Remez-Type Inequalities

“…i is a universal polynomial of degree n − 1 in 1 ǫ whose coefficients are positive numbers related to Vitushkin's bounds for covering numbers of polynomial sub-level sets of degree d (see [37,39,23]). The explicit formula for M n,d is given in [39]. In particular,…”

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The classical Remez inequality ([33]) bounds the maximum of the absolute value of a real polynomial P of degree d on [−1, 1] through the maximum of its absolute value on any subset Z ⊂ [−1, 1] of positive Lebesgue measure. Extensions to several variables and to certain sets of Lebesgue measure zero, massive in a much weaker sense, are available (see, e.g., [14,39,8]-norming sets are exactly those not contained in any algebraic hypersurface of degree d in R n , there are many apparently unrelated reasons for Z ⊂ [−1, 1] n to have this property. )In the present paper we study norming sets and related Remez-type inequalities in a general setting of finite-dimensional linear spaces V of continuous functions on [−1, 1] n , remaining in most of the examples in the classical framework.

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Remez-Type Inequality for Smooth Functions

“…In what follows we shall omit the dimension n from the notations for ω d (Z) = ω n,d (Z). It was shown in [12] that in many cases (but not always!) the bound of Proposition 2.1 is pretty sharp.…”

“…The quantity ω n,d (Z) can be effectively estimated in terms of the metric entropy of Z and it may be nonzero for discrete and even finite sets Z.In the present paper we extend Remez inequality to functions of finite smoothness. This is done by combining the result of [12] with the Taylor polynomial approximation of smooth functions. As a consequence we obtain explicit lower bounds in some examples in the Whitney problem of a C k -smooth extrapolation from a given set Z, in terms of the geometry of Z.----------------…”

A case of multivariate Birkhoff interpolation using high order derivatives

“…The classical Remez inequality and its generalizations compare maxima of a polynomial P on two sets U ⊂ G (see [6,5,21,25] and references therein). We would like to extend this setting, in order to include into sampling information on U the derivatives of P .…”