Nonuniform sampling and approximation in Sobolev space from perturbation of the framelet system  

作　　者：Youfa Li Deguang Han Shouzhi Yang Ganji Huang 

机构地区：[1]College of Mathematics and Information Science,Guangxi University,Nanning 530004,China [2]Department of Mathematics,University of Central Florida,Orlando,FL 32816,USA [3]Department of Mathematics,University of Shantou,Shantou 535063,China

出　　处：《Science China Mathematics》2021年第2期351-372,共22页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.61961003,61561006 and 11501132);Natural Science Foundation of Guangxi(Grant Nos.2018JJA110110 and 2016GXNSFAA380049);the talent project of Education Department of Guangxi Government for Young-Middle-Aged Backbone Teachers;supported by National Science Foundation of USA(Grant No.DMS-1712602)。

摘　　要：The Sobolev space H^(■)(R^(d)),where■>d/2,is an important function space that has many applications in various areas of research.Attributed to the inertia of a measurement instrument,it is desirable in sampling theory to recover a function by its nonuniform sampling.In the present paper,based on dual framelet systems for the Sobolev space pair(H^(s)(R^(d)),H^(-s)(R^(d))),where d/2<s<■,we investigate the problem of constructing the approximations to all the functions in H^(■)(R^(d))by nonuniform sampling.We first establish the convergence rate of the framelet series in(H^(s)(R^(d)),H^(-s)(R^(d))),and then construct the framelet approximation operator that acts on the entire space H^(■)(R^(d)).We examine the stability property for the framelet approximation operator with respect to the perturbations of shift parameters,and obtain an estimate bound for the perturbation error.Our result shows that under the condition d/2<s<■,the approximation operator is robust to shift perturbations.Motivated by Hamm(2015)’s work on nonuniform sampling and approximation in the Sobolev space,we do not require the perturbation sequence to be in■^(α)(Z^(d)).Our results allow us to establish the approximation for every function in H^(■)(R^(d))by nonuniform sampling.In particular,the approximation error is robust to the jittering of the samples.

关 键 词：Sobolev space framelet series truncation error perturbation error nonuniform sampling and approximation 

分 类 号：TN911.6[电子电信—通信与信息系统]