NORMING SETS AND RELATED REMEZ-TYPE INEQUALITIES

Abstract

The classical Remez inequality [‘Sur une propriété des polynomes de Tchebycheff’,Comm. Inst. Sci. Kharkov13(1936), 9–95] 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$Z\subset [-1,1]$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, for example, Brudnyi and Ganzburg [‘On an extremal problem for polynomials of$n$variables’,Math. USSR Izv.37(1973), 344–355], Yomdin [‘Remez-type inequality for discrete sets’,Israel. J. Math.186(2011), 45–60], Brudnyi [‘On covering numbers of sublevel sets of analytic functions’,J. Approx. Theory162(2010), 72–93]). Still, given a subset$Z\subset [-1,1]^{n}\subset \mathbb{R}^{n}$, it is not easy to determine whether it is${\mathcal{P}}_{d}(\mathbb{R}^{n})$-norming (here${\mathcal{P}}_{d}(\mathbb{R}^{n})$is the space of real polynomials of degree at most$d$on$\mathbb{R}^{n}$), that is, satisfies a Remez-type inequality:$\sup _{[-1,1]^{n}}|P|\leq C\sup _{Z}|P|$for all$P\in {\mathcal{P}}_{d}(\mathbb{R}^{n})$with$C$independent of$P$. (Although${\mathcal{P}}_{d}(\mathbb{R}^{n})$-norming sets are precisely those not contained in any algebraic hypersurface of degree$d$in$\mathbb{R}^{n}$, there are many apparently unrelated reasons for$Z\subset [-1,1]^{n}$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 nontrivial examples. Next, we extend the Turán–Nazarov inequality for exponential polynomials to several variables, and on this basis 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.