您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

《山东大学学报(理学版)》 ›› 2026, Vol. 61 ›› Issue (2): 50-57.doi: 10.6040/j.issn.1671-9352.0.2024.410

• • 上一篇    

平均框架下多元逼近问题的(s,t)-弱可处理性

陈佳1,燕慧超2,刘有军1   

  1. 1.山西大同大学数学与统计学院, 山西 大同 037009;2.山西大同大学计算机与网络工程学院, 山西 大同 037009
  • 发布日期:2026-02-13
  • 作者简介:陈佳(1987— ),女,副教授,博士,研究方向为函数逼近论. E-mail:jiachencd@163.com
  • 基金资助:
    国家自然科学基金资助项目(12001342);山西省高等学校科技创新项目(2022L438);山西大同大学基础青年科研基金项目(2022Q10);山西大同大学博士科研启动经费项目(2021-B-17,2019-B-10)

(s,t)-weak tractability of multivariate approximation problems in the average case setting

CHEN Jia1, YAN Huichao2, LIU Youjun1   

  1. 1. School of Mathematics and Statistics Science, Shanxi Datong University, Datong 037009, Shanxi, China;
    2. School of Computer and Network Engineering, Shanxi Datong University, Datong 037009, Shanxi, China
  • Published:2026-02-13

PDF (PC)

9

摘要： 研究在平均框架下具有零平均高斯测度的Banach空间中的多元逼近问题APPd(d∈N+),其中,零平均高斯测度的协方差核具有非负权重序列{αj}和{γj},特别地,介绍两类具有不同权重的协方差核。利用有限个连续线性泛函构成的算法来逼近多元问题APPd。讨论在绝对误差和归一误差下,这两类具有不同权重的协方差核的Banach空间中的L2逼近问题APP={APPd}d∈N+的(s,t)-弱可处理性(s>0, t≥1)。最后,利用实分析方法得到这两类L2逼近问题APP是(s,1)-弱可处理的充分且必要条件:当j趋于无穷大时,权重序列{γj}的极限为0。

关键词: L2逼近, (s,t)-弱可处理性, 平均框架, 协方差核

Abstract: This paper investigated multivariate approximation problems APPd(d∈N+)of Banach spaces equipped with zero-mean Gaussian measures in the average case setting, where covariance kernels of the zero-mean Gaussian measures had non-negative weighted sequences {αj} and {γj}. In particular, the paper introduced covariance kernels with two different weights. We approximated the multivariate problems APPd by the algorithms that used finitely many continuous linear functionals. This paper discussed(s,t)-weak tractability for s>0 and t≥1 of the L2-approximation problems APP={APPd}d∈N+ from the Banach spaces with the above two weighted covariance kernels under the absolute error criterion and the normalized error criterion. As a result, by the real analyzing the sufficient and necessary condition for(s,1)-weak tractability of these two L2-approximation problems APP could be obtained as follows: the weight sequence {γj} tendde to 0 as j tends to infinity.

Key words: L2-approximation, (s,t)-weak tractability, average case setting, covariance kernels

中图分类号: 

  • O174
[1] BERLINET A,THOMAS-AGNAN C. Reproducing kernel Hilbert spaces in probability and statistics[M]. New York: Springer, 2004:75-94.
[2] GLIMM J, JAFFE A. Quantum physics[M]. New York: Springer, 1987:273-471.
[3] TRAUB J F, WERSCHULZ A G. Complexity and information[M]. Cambridge: Cambridge University Press, 1998:43-52.
[4] NOVAK E, WOZNIAKOWSKI H. Tractability of multivariate problems, volume I: linear information[M]. Zürich: European Mathematical Society, 2008:14-283.
[5] NOVAK E, WOZNIAKOWSKI H. Tractability of multivariate problems, volume II: standard information for functionals[M]. Zürich: European Mathematical Society, 2010:71-296.
[6] NOVAK E, WOZNIAKOWSKI H. Tractability of multivariate problems, volume III: standard information for operators[M]. Zürich:European Mathematical Society, 2012:185-440.
[7] LIU Yongping, XU Guiqiao. Average case tractability of a multivariate approximation problem[J]. Journal of Complexity, 2017, 43:76-102.
[8] LIU Yongping, XU Guiqiao.(s,t)-weak tractability of multivariate linear problems in the average case setting[J]. Acta Mathematica Scientia, 2019, 39(4):1033-1052.
[9] CHEN Jia, WANG Heping, ZHANG Jie. Average case (s,t)-weak tractability of non-homogeneous tensor product problems[J]. Journal of Complexity, 2018, 49:27-45.
[10] FASSHAUER G E, HICKERNELL F J, WOZNIAKOWSKI H. Average case approximation: convergence and tractability of Gaussian kernels[M] //PLASKOTA L, WOZNIAKOWSKI H, et al. Monte Carlo and Quasi-Monte Carlo 2010. Berlin:Springer, 2012:329-345.
[11] KHARTOV A A. A simplified criterion for quasi-polynomial tractability of approximation of random elements and its applications[J]. Journal of Complexity, 2016, 34:30-41.
[12] CHEN Jia, WANG Heping. Average case tractability of multivariate approximation with Gaussian kernels[J]. Journal of Approximation Theory, 2019, 239:51-71.
[13] LIFSHITS M A, PAPAGEORGIOU A, WOZNIAKOWSKI H. Average case tractability of non-homogeneous tensor product problems[J]. Journal of Complexity, 2012, 28:539-561.
[14] XU Guiqiao. Quasi-polynomial tractability of linear problems in the average case setting[J]. Journal of Complexity, 2014, 30:54-68.
[15] XU Guiqiao. Tractability of linear problems defined over Hilbert spaces[J]. Journal of Complexity, 2014, 30:735-749.
[16] YAN Huichao, CHEN Jia. Tractability of approximation of functions defined over weighted Hilbert spaces[J]. Axioms, 2024, 13:108.
[1] 常佳乐,苏先锋. 一类复微分差分多项式的零点分布[J]. 《山东大学学报(理学版)》, 2024, 59(8): 28-33.
[2] 刘小玉,魏明权,宋鹏超. 高维乘积空间上反向分数次Hardy不等式的最佳常数[J]. 《山东大学学报(理学版)》, 2024, 59(8): 20-27.
[3] 刘占宏,陶双平. 分数次极大算子及交换子在Morrey空间上的加权估计[J]. 《山东大学学报(理学版)》, 2024, 59(6): 108-115.
[4] 张伟. 连续广义框架的ε-近似[J]. 《山东大学学报(理学版)》, 2024, 59(4): 73-80.
[5] 芮俪,逯光辉,李雪梅. 广义加权变指标Morrey空间上双线性θ-型Calderón-Zygmund算子[J]. 《山东大学学报(理学版)》, 2024, 59(4): 62-72.
[6] 赵明馨,孙桂荣,黄志刚. Fermat型微分-差分方程的整函数解[J]. 《山东大学学报(理学版)》, 2023, 58(6): 107-112.
[7] 李菲,杨刘. 关于全纯曲线的Milloux型不等式[J]. 《山东大学学报(理学版)》, 2023, 58(6): 99-106.
[8] 古勇毅,孔荫莹. 一类辅助微分方程的亚纯解及其应用[J]. 《山东大学学报(理学版)》, 2019, 54(8): 81-89.
[9] 张伟,付艳玲. Hilbert空间中连续广义框架的分解[J]. 《山东大学学报(理学版)》, 2019, 54(8): 76-80.
[10] 陶双平,杨雨荷. 分数次极大算子及交换子在λ-中心Morrey空间上的加权估计[J]. 《山东大学学报(理学版)》, 2019, 54(8): 68-75.
[11] 张伟杰,王新利,王汉杰. 亚纯函数微分多项式IM分担值的唯一性[J]. 《山东大学学报(理学版)》, 2019, 54(8): 90-96.
[12] 赵欢,周疆. 变指数Herz型Hardy空间上的多线性Calderón-Zygmund算子交换子[J]. 山东大学学报（理学版）, 2018, 53(10): 42-50.
[13] 盖晓华,郭学军,冯金顺,陈清江,程正兴. 高维小波框架包子空间对空间L2(Rn)的分解[J]. 山东大学学报（理学版）, 2018, 53(8): 34-42.
[14] 张石梅,吴秀碧,龙见仁. 复线性微分方程解的增长性的进一步讨论[J]. 山东大学学报（理学版）, 2018, 53(6): 23-29.
[15] 陶双平,高荣. 多线性分数次积分和极大算子在Morrey空间上的加权估计[J]. 山东大学学报（理学版）, 2018, 53(6): 30-37.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发