Norming sets and related Remez-type inequalities (Q2800018)

In the theory of polynomial inequalities, a well known result by \textit{E. J. Remez} [``Sur une propiété des polynômes de Tchebycheff'', Comm. Inst. Sci. Kharkov 13, 93--95 (1936)], gives a bound of the maximum of the absolute value of an univariate real polynomial of degree \(m\) on the interval \([-1,1]\) in terms of the maximum of its absolute value on any subset \(Z\subset[-1,1]\) of positive Lebesgue measure \(\mu(Z)\). Indeed, sup \(\{|p(x)|, x\in [-1,1]\} \leq T_{m} (\frac{4-\mu(Z)}{\mu(Z)})\sup \{|p(x)|, x\in Z\}\), where \(T_{m}(x)\) is the Chebyshev polynomial of first kind and degree \(m\). For the multidimensional case, in [\textit{Yu. A. Brudnyj} and \textit{M. I. Ganzburg}, Izv. Akad. Nauk SSSR, Ser. Mat. 37, 344--355 (1973; Zbl 0283.26012)], the authors deduce the following Remez-type inequality: if \(B\subset \mathbb R^{n}\) is a convex body and \(Z\subset B\) is a measurable subset of \(B\), then for every polynomial \(p\) in \(n\) variables and degree \(m\), you get \(\sup \{|p(x)|: x\in B\} \leq T_m(\frac{1+(1-\lambda)^{1/ n}}{1 - (1-\lambda)^{1/n}})\sup \{|p(x)|: x\in Z\}\). Here, \(\lambda= \frac{\mu_{n}(Z)}{\mu_{n}(B)},\) and \(\mu_{n}\) denotes the Lebesgue measure in \(\mathbb R^{n}\). Inequalities as above are also true for some sets of zero Lebesgue measure and also for some finite sets.NEWLINENEWLINENEWLINEIn the paper under review, the authors deal with the analysis of norming sets and related Remez-type inequalities in a general setting of finite dimensional linear spaces of continuous functions on closed unit hypercubes. Let \(V\subset C(Q_{1}^{n})\) be a finite-dimensional subspace of real continuous functions on the closed unit cube \(Q_{1}^{n}= [0,1]^{n}\). A compact subset \(Z\subset Q_{1}^{n}\) is said to be \(V\)-norming if there exists a constant \(C>0\) such that, for every \(f\in V,\) \(\max_{Q_{1}^{n}} |f| \leq C\max_{Z} |f|\). The \(V\)-norming constant of \(Z,\) denoted by \(N_{V}(Z),\) is the minimum of all such constants \(C\).NEWLINENEWLINENEWLINEFirst, some sufficient conditions for \(Z\) to be \(V\)-norming when \(V\) is the space of real polynomials in \(n\) variables of degree at most \(d\) are given in different frameworks: interpolation systems, sets with algebraically independent coordinates, nodal sets of elliptic partial differential equations, algebraic curves of higher degree, transcendental surfaces by using different tools like potential theory, Haussdorf measure and metric entropy, algebraic geometry, among others. In such contexts, estimates of \(N_{V}(Z)\) are given.NEWLINENEWLINENEWLINENext, dealing with exponential polynomials of several variables \(p(x)= \sum_{k=0}^{m} c_{k} \exp f_{k}(x)\) where \(f_{k}\) are complex-valued linear functionals on \(\mathbb R^{n}\), an extension of the Turàn-Nazarov inequality is given for a convex body \(B\) contained in a \(d\)-dimensional affine subspace \(A\) in \(\mathbb R^{n}\) and a Borel subset \(Z\) in \(B\) (Theorem 3.2). As a consequence, a ``fewnomial'' Remez-type inequality (Theorem 3.3) is given for \(d\)-dimensional logarithmically convex compact subsets in \(d\)-dimensional affine submanifolds in \((\mathbb R^{*}_{+})^{n}\) and \(p(x)= \sum_{k=0}^{m} c_{k} x^{\alpha_{k}}\), where \(x\in(\mathbb R^{*}_{+})^{n}\) and \(x^{\beta}\) denotes the monomial associated with \(\beta = (\beta_{1}, \ldots, \beta_{n})\).NEWLINENEWLINENEWLINEFinally, it is proved that the best constant \(N_{V}(Z)\) in the Remez-type inequality satisfies a Lipschitz continuity property in terms of \(Z\) with respect to the Haussdorf metric. As a consequence, the norming property of a compact set is an open condition in the Haussdorf metric, with an explicit bound on the norming constant of nearby sets.