This article considers linear approximation based on function evaluations in reproducing kernel Hilbert spaces of the Gaussian kernel and a more general class of weighted power series kernels on the interval $[-1, 1]$. We derive almost matching upper and lower bounds on the worst-case error, measured both in the uniform and $L^2([-1,1])$-norm, in these spaces. The results show that if the power series kernel expansion coefficients $\alpha_n^{-1}$ decay at least factorially, their rate of decay controls that of the worst-case error. Specifically, (i) the $n$th minimal error decays as $\alpha_n^{{ -1/2}}$ up to a sub-exponential factor and (ii) for any $n$ sampling points in $[-1, 1]$ there exists a linear algorithm whose error is $\alpha_n^{{ -1/2}}$ up to an exponential factor. For the Gaussian kernel the dominating factor in the bounds is $(n!)^{-1/2}$.
翻译:本文根据复制 Gausian 内核内核的内核Hilbert 空间的功能评价以及间隔 $[1,1] 的更一般的加权电源序列内核,考虑了线性近似值。我们从最坏的错误中得出几乎匹配的上限和下限,以制服和$[2,[1,1,1]美元-诺尔米衡量,这些空间。结果显示,如果电源序列内核扩张系数$\alpha_n ⁇ _ ⁇ 1,1美元至少以因数计衰减,它们的衰减率控制在最坏的错误中。具体地说,(一) 美元的最低误差以 $\alpha_n\\\\\\\\1/2美元递增至亚特因数计算,以及(二) $1,则存在一条线性算法,其误差为$\alpha_n\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\