平均框架下多元逼近问题的(s,t)-弱可处理性

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

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APP_d(d∈N₊),其中,零平均高斯测度的协方差核具有非负权重序列{α_j}和{γ_j},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APP_d。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={APP_d}_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{γ_j}的极限为0。

Abstract

This paper investigated multivariate approximation problems APP_d(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 {α_j} and {γ_j}. In particular, the paper introduced covariance kernels with two different weights. We approximated the multivariate problems APP_d 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={APP_d}_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 {γ_j} tend to 0 as j tends to infinity.

关键词

L₂逼近 / (s,t)-弱可处理性 / 平均框架 / 协方差核

Key words

L₂-approximation / (s,t)-weak tractability / average case setting / covariance kernels

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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)

山西大同大学博士科研启动经费项目(2019-B-10)