NORMING SETS AND RELATED REMEZ-TYPE INEQUALITIES

The classical Remez inequality [‘Sur une propriété des polynomes de Tchebycheff’, Comm. Inst. Sci. Kharkov 13 (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. Theory 162 (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.

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