Average sampling numbers of multivariate periodic function spaces with a Gaussian measure

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• 作者：Zexia Huang^a ; ^{xhu_hzx@163.com" class="auth_mail" title="E-mail the corresponding author} ; Heping Wang^b ; ^{wanghp@cnu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Average sampling numbers ; Average approximation numbers ; Asymptotically optimal ; Gaussian measure
• 刊名：Journal of Complexity
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：36
• 期：Complete
• 页码：31-45
• 全文大小：423 K

文摘

In this paper, we study average sampling numbers of the multivariate periodic function space ${\mathring{L}}_2$ with a Gaussian measure μ$\mu$ in the L_q$L_q$ metric for 1≤q≤∞$1\le q\le \infty$, and obtain their asymptotical orders, where the Cameron–Martin space of the measure μ$\mu$ is an anisotropic periodic Sobolev space. Moreover, we show that in the average case setting, the Lagrange interpolating operators are asymptotically optimal linear algorithms in the L_q$L_q$ metric for all 1≤q≤∞$1\le q\le \infty$. This is different from the situation in the worst case setting, where the Lagrange interpolating operators are not asymptotically optimal linear algorithms in the L_q$L_q$ metric for q=1$q=1$ or ∞$\infty$.