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On Generalizations of p-Sets and their Applications
Zhou, Heng1; Xu, Zhiqiang2,3
AbstractThe p-set, which is in a simple analytic form, is well distributed in unit cubes. The well-known Weil's exponential sum theorem presents an upper bound of the exponential sum over the p-set. Based on the result, one shows that the p-set performs well in numerical integration, in compressed sensing as well as in uncertainty quantification. However, p-set is somewhat rigid since the cardinality of the p-set is a prime p and the set only depends on the prime number p. The purpose of this paper is to present generalizations of p-sets, say P-d,p(a,epsilon) which is more flexible. Particularly, when a prime number p is given, we have many different choices of the new p-sets. Under the assumption that Goldbach conjecture holds, for any even number m, we present a point set, say L-p,L-q, with cardinality m - 1 by combining two different new p-sets, which overcomes a major bottleneck of the p-set. We also present the upper bounds of the exponential sums over P-d,p(a,epsilon) and L-p,L-q, which imply these sets have many potential applications.
Keywordp-set deterministic sampling numerical integral exponential sum
Funding ProjectNSFC[91630203] ; NSFC[11331012] ; NSFC[11688101] ; NSFC[61602341] ; National Basic Research Program of China (973 Program)[2015CB856000]
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000453759200006
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Document Type期刊论文
Corresponding AuthorZhou, Heng
Affiliation1.Tianjin Polytech Univ, Sch Sci, Tianjin 300160, Peoples R China
2.Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math, LSEC, Beijing 100190, Peoples R China
3.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
Recommended Citation
GB/T 7714
Zhou, Heng,Xu, Zhiqiang. On Generalizations of p-Sets and their Applications[J]. NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS,2019,12(2):453-466.
APA Zhou, Heng,&Xu, Zhiqiang.(2019).On Generalizations of p-Sets and their Applications.NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS,12(2),453-466.
MLA Zhou, Heng,et al."On Generalizations of p-Sets and their Applications".NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS 12.2(2019):453-466.
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