On the cardinality of lower sets and universal discretization

IF 1.8 2区数学 Q1 MATHEMATICS

Journal of Complexity Pub Date : 2023-06-01 DOI:10.1016/j.jco.2022.101726

F. Dai , A. Prymak , A. Shadrin , V.N. Temlyakov , S. Tikhonov

引用次数: 0

Abstract

A set Q in $Z_{+}^{d}$ is a lower set if $(k_{1}, \dots, k_{d}) \in Q$ implies $(l_{1}, \dots, l_{d}) \in Q$ whenever $0 \leq l_{i} \leq k_{i}$ for all i. We derive new and refine known results regarding the cardinality of the lower sets of size n in $Z_{+}^{d}$ . Next we apply these results for universal discretization of the $L_{2}$ -norm of elements from n-dimensional subspaces of trigonometric polynomials generated by lower sets.

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下集的基数性与泛离散化

如果（k1，…，kd）∈Q对所有i意味着（l1，…，ld）∈Q，只要0≤li≤ki。我们导出了关于Z+d中大小为n的较低集合的基数的新的和改进的已知结果。接下来，我们将这些结果应用于由较低集合生成的三角多项式的n维子空间的元素的L2范数的通用离散化。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Complexity 工程技术-计算机：理论方法

CiteScore

3.10

自引率

17.60%

发文量

审稿时长

>12 weeks

期刊介绍： The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited. Areas Include: • Approximation theory • Biomedical computing • Compressed computing and sensing • Computational finance • Computational number theory • Computational stochastics • Control theory • Cryptography • Design of experiments • Differential equations • Discrete problems • Distributed and parallel computation • High and infinite-dimensional problems • Information-based complexity • Inverse and ill-posed problems • Machine learning • Markov chain Monte Carlo • Monte Carlo and quasi-Monte Carlo • Multivariate integration and approximation • Noisy data • Nonlinear and algebraic equations • Numerical analysis • Operator equations • Optimization • Quantum computing • Scientific computation • Tractability of multivariate problems • Vision and image understanding.