On Discretizing Uniform Norms of Exponential Sums

Abstract

AbstractIn this paper, we consider the uniform norm discretization problem for general real multivariate exponential sums $$p({{\mathbf{w}}})=\sum _{0\le j\le n}c_je^{\langle \mu _j, {{\mathbf{w}}}\rangle }, \;\;\mu _j, {{\mathbf{w}}}\in \mathbb {R}^d$$ p ( w ) = ∑ 0 ≤ j ≤ n c j e ⟨ μ j , w ⟩ , μ j , w ∈ R d . Given arbitrary $$0<\tau \le 1$$ 0 < τ ≤ 1 this problem consists in finding discrete point sets $$ {{\mathbf{w}}}_j\in K, 1\le j\le N$$ w j ∈ K , 1 ≤ j ≤ N in the compact domain $$K\subset \mathbb {R}^d, d\ge 1$$ K ⊂ R d , d ≥ 1 so that for every $$p({{\mathbf{w}}})$$ p ( w ) as above we have $$\begin{aligned} \max _{{{\mathbf{w}}}\in K}|p({{\mathbf{w}}})|\le (1+\tau )\max _{1\le j\le N}|p({{\mathbf{w}}}_j)|. \end{aligned}$$ max w ∈ K | p ( w ) | ≤ ( 1 + τ ) max 1 ≤ j ≤ N | p ( w j ) | . Using certain new Bernstein–Markov type inequalities for exponential sums it will be verified that for convex polytopes and convex polyhedral cones K in $$\mathbb {R}^d, d\ge 1$$ R d , d ≥ 1 there exist meshes $${{\mathbf{w}}}_1,\ldots ,{{\mathbf{w}}}_N\subset K$$ w 1 , … , w N ⊂ K of cardinality $$\begin{aligned} N\le c\left( \frac{n}{\sqrt{\tau }}\right) ^{d}\ln ^{d}\frac{\mu _n^*}{\delta \tau }, \;\;\;\mu _n^*:=\max _{1\le j\le n}|\mu _j| \end{aligned}$$ N ≤ c n τ d ln d μ n ∗ δ τ , μ n ∗ : = max 1 ≤ j ≤ n | μ j | for which the above inequality holds for any multivariate exponential sum p with exponents satisfying the separation condition $$|\mu _{k}-\mu _j|\ge \delta , j\ne k, \delta >0$$ | μ k - μ j | ≥ δ , j ≠ k , δ > 0 . In addition, the optimality of the cardinality estimates will be also discussed.