F. Dai, A. Prymak, A. Shadrin, V. Temlyakov, S. Tikhonov
{"title":"On cardinality of the lower sets and universal discretization","authors":"F. Dai, A. Prymak, A. Shadrin, V. Temlyakov, S. Tikhonov","doi":"10.48550/arXiv.2208.02113","DOIUrl":null,"url":null,"abstract":"A set $Q$ in $\\mathbb{Z}_+^d$ is a lower set if $(k_1,\\dots,k_d)\\in Q$ implies $(l_1,\\dots,l_d)\\in Q$ whenever $0\\le l_i\\le k_i$ for all $i$. We derive new and refine known results regarding the cardinality of the lower sets of size $n$ in $\\mathbb{Z}_+^d$. 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.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":"18 1","pages":"101726"},"PeriodicalIF":2.2000,"publicationDate":"2022-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of complex networks","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.48550/arXiv.2208.02113","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 1
Abstract
A set $Q$ in $\mathbb{Z}_+^d$ is a lower set if $(k_1,\dots,k_d)\in Q$ implies $(l_1,\dots,l_d)\in Q$ whenever $0\le l_i\le k_i$ for all $i$. We derive new and refine known results regarding the cardinality of the lower sets of size $n$ in $\mathbb{Z}_+^d$. 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.
期刊介绍：
Journal of Complex Networks publishes original articles and reviews with a significant contribution to the analysis and understanding of complex networks and its applications in diverse fields. Complex networks are loosely defined as networks with nontrivial topology and dynamics, which appear as the skeletons of complex systems in the real-world. The journal covers everything from the basic mathematical, physical and computational principles needed for studying complex networks to their applications leading to predictive models in molecular, biological, ecological, informational, engineering, social, technological and other systems. It includes, but is not limited to, the following topics: - Mathematical and numerical analysis of networks - Network theory and computer sciences - Structural analysis of networks - Dynamics on networks - Physical models on networks - Networks and epidemiology - Social, socio-economic and political networks - Ecological networks - Technological and infrastructural networks - Brain and tissue networks - Biological and molecular networks - Spatial networks - Techno-social networks i.e. online social networks, social networking sites, social media - Other applications of networks - Evolving networks - Multilayer networks - Game theory on networks - Biomedicine related networks - Animal social networks - Climate networks - Cognitive, language and informational network
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
