Norming Sets and Related Remez-Type Inequalities

Abstract

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. First, we discuss some sufficient conditions for Z to be V -norming, partly known, partly new, restricting ourselves to the simplest non-trivial examples. Next, we extend the Turan-Nazarov inequality for exponential polynomials to several variables, and on this base prove a new fewnomial Remez-type inequality. Finally, we study the family of optimal constants N V (Z) in the Remez-type inequalities for V , as the function of the set Z, showing that it is Lipschitz in the Hausdorff metric.

Lecture notes on quantitative unique continuation for solutions of second order elliptic equations

“…There are interesting generalizations of the Remez inequality, in particular the measure of the set can be replaced by another geometric characteristic; higher dimensional version are also known, we refer the reader to [5,12].…”

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 .…”

“…The second question is to provide explicit estimates of the robustness of a certain polynomial interpolation (or reconstruction) scheme. This leads to "Remez-type" (or "Norming") inequalities (see [5,10] and references therein). Our second main result is a "Birkhoff-Remez" inequality for the interpolation scheme.…”

Smooth parametrizations in dynamics, analysis, diophantine and computational geometry

“…. , s. It is well known that W is contained in an algebraic hypersurface of degree less than or equal to d if and only if the rank of A is less than τ (see, for example, [10], Lemma 1,or [16], Proposition 2.2). Now the idea is to apply Lemma 5.1 to the smooth mapping ψ = V d • φ : I n → R τ .…”

“…Inequalities of the form (5.5) are known also for sets Z of measure zero, for discrete or finite Z (see [16,82,83] and references therein). Similar inequalities have been studied for restrictions of polynomials to semi-algebraic (subanalytic) sets ([2]- [4], [11,11,13,15,23,24,53,83]).…”