On Discretizing Uniform Norms of Exponential Sums

Abstract

In this paper, we consider the uniform norm discretization problem for general real multivariate exponential sums p({{mathbf{w}}})=sum _{0le jle n}c_je^{langle mu _j, {{mathbf{w}}}rangle }, ;;mu _j, {{mathbf{w}}}in mathbb {R}^d. Given arbitrary 0<tau le 1 this problem consists in finding discrete point sets {{mathbf{w}}}_jin K, 1le jle N in the compact domain Ksubset mathbb {R}^d, dge 1 so that for every p({{mathbf{w}}}) as above we have maxw∈K|p(w)|≤(1+τ)max1≤j≤N|p(wj)|.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\max _{{{\\mathbf{w}}}\\in K}|p({{\\mathbf{w}}})|\\le (1+\ au )\\max _{1\\le j\\le N}|p({{\\mathbf{w}}}_j)|. \\end{aligned}$$\\end{document}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, dge 1 there exist meshes {{mathbf{w}}}_1,ldots ,{{mathbf{w}}}_Nsubset K of cardinality N≤cnτdlndμn∗δτ,μn∗:=max1≤j≤n|μj|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} N\\le c\\left( \\frac{n}{\\sqrt{\ au }}\\right) ^{d}\\ln ^{d}\\frac{\\mu _n^*}{\\delta \ au }, \\;\\;\\;\\mu _n^*:=\\max _{1\\le j\\le n}|\\mu _j| \\end{aligned}$$\\end{document}for which the above inequality holds for any multivariate exponential sum p with exponents satisfying the separation condition |mu _{k}-mu _j|ge delta , jne k, delta >0. In addition, the optimality of the cardinality estimates will be also discussed.