论文标题
$ l_p $ - $ l_ \ infty $的非压缩子类采样恢复
$L_p$-Sampling recovery for non-compact subclasses of $L_\infty$
论文作者
论文摘要
在本文中，我们研究了某些相关的多元功能类的采样恢复问题，这些函数类别不会紧凑到$ l_ \ infty $中。因此，在统一规范中与Kolmogorov宽度有关的最新工具不适用。从某种意义上说，我们通过在BESOV和Triebel-Lizorkin空间的背景下考虑“非常小”的平滑度来继续研究小小的平滑度问题，并以主导的规律性为主。除了奥斯瓦尔德在单变量情况下，恢复此类功能的恢复还不多。作为第一步，我们证明了固定级别的Faber-Schauder系数的$ \ ell_p $ -norm的统一界限。使用此功能，我们能够控制（Smolyak）截断的Faber-Schauder系列中的错误，其中$ Q <\ infty $。事实证明，主收敛的主要速率很清晰。结果，我们也获得了$ s^1_ {1，\ infty} f（[0,1]^d）$的结果，一个空间``close'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''（[0,1]^d）$，在数值分析中很重要，尤其是数值集成，尤其是较差的傅立叶分析属性。
In this paper we study the sampling recovery problem for certain relevant multivariate function classes which are not compactly embedded into $L_\infty$. Recent tools relating the sampling numbers to the Kolmogorov widths in the uniform norm are therefore not applicable. In a sense, we continue the research on the small smoothness problem by considering "very" small smoothness in the context of Besov and Triebel-Lizorkin spaces with dominating mixed regularity. There is not much known on the recovery of such functions except of an old result by Oswald in the univariate situation. As a first step we prove the uniform boundedness of the $\ell_p$-norm of the Faber-Schauder coefficients in a fixed level. Using this we are able to control the error made by a (Smolyak) truncated Faber-Schauder series in $L_q$ with $q<\infty$. It turns out that the main rate of convergence is sharp. As a consequence we obtain results also for $S^1_{1,\infty}F([0,1]^d)$, a space which is ``close'' to the space $S^1_1W([0,1]^d)$ which is important in numerical analysis, especially numerical integration, but has rather bad Fourier analytic properties.
