On Discretizing Uniform Norms of Exponential Sums
نویسندگان
چکیده
Abstract In this paper, we consider the uniform norm discretization problem for general real multivariate exponential sums $$p({{\mathbf{w}}})=\sum _{0\le j\le n}c_je^{\langle \mu _j, {{\mathbf{w}}}\rangle }, \;\;\mu {{\mathbf{w}}}\in \mathbb {R}^d$$ p ( w ) = ∑ 0 ≤ j n c e ⟨ μ , ⟩ ∈ R d . Given arbitrary $$0<\tau \le 1$$ < τ 1 consists in finding discrete point sets $$ {{\mathbf{w}}}_j\in K, 1\le N$$ K N compact domain $$K\subset {R}^d, d\ge ⊂ ≥ so that every $$p({{\mathbf{w}}})$$ as above have $$\begin{aligned} \max _{{{\mathbf{w}}}\in K}|p({{\mathbf{w}}})|\le (1+\tau )\max _{1\le N}|p({{\mathbf{w}}}_j)|. \end{aligned}$$ max | + . Using certain new Bernstein–Markov type inequalities it will be verified convex polytopes and polyhedral cones K $$\mathbb there exist meshes $${{\mathbf{w}}}_1,\ldots ,{{\mathbf{w}}}_N\subset K$$ … of cardinality N\le c\left( \frac{n}{\sqrt{\tau }}\right) ^{d}\ln ^{d}\frac{\mu _n^*}{\delta \tau \;\;\;\mu _n^*:=\max n}|\mu _j| ln ∗ δ : which inequality holds any sum p with exponents satisfying separation condition $$|\mu _{k}-\mu _j|\ge \delta , j\ne k, >0$$ k - ≠ > addition, optimality estimates also discussed.
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ژورنال
عنوان ژورنال: Constructive Approximation
سال: 2022
ISSN: ['0176-4276', '1432-0940']
DOI: https://doi.org/10.1007/s00365-022-09565-6