It was recently shown by D. Krieg and M. Ullrich that, for $L_2$-approximation of functions from a reproducing kernel Hilbert space, function values are almost as powerful as arbitrary linear information if the approximation numbers are square-summable. That is, $e_n\lesssim \sqrt{\frac{1}{k_n}\sum _{j\ge k_n}{a}_j^2}\text{with}{k}_n\asymp \frac{n}{ln\left(n\right)},$where $e_n$ are the sampling numbers and $a_k$ are the approximation numbers. In particular, if $\left(a_k\right)\in {\ell }_2$, then $e_n$ and $a_n$ are of the same polynomial order. For this, we presented an explicit (weighted least squares) algorithm based on i.i.d. random points and proved that this works with positive probability. This implies the existence of a good deterministic sampling algorithm.

Here, we present a modification of our proof that shows that the same algorithm works with probability at least $1-n^{-c}$ for any given $c>0$.

D. Krieg和M. Ullrich最近证明， ${\mathrm{大号}}_2$从可再生内核Hilbert空间对函数进行逼近，如果逼近数是平方和，则函数值几乎与任意线性信息一样强大。那是，${Ë}_{ñ}\lesssim \sqrt{\frac{\mathrm{1个}}{{ķ}_{ñ}}\sum _{Ĵ\ge {ķ}_{ñ}}{\mathrm{一种}}_{Ĵ}^2}\text{与}{ķ}_{ñ}\asymp \frac{ñ}{ln（ñ）}，$哪里 ${Ë}_{ñ}$ 是采样数和 ${\mathrm{一种}}_{ķ}$是近似数。特别是如果$（{\mathrm{一种}}_{ķ}）\in {\ell }_2$， 然后 ${Ë}_{ñ}$ 和 ${\mathrm{一种}}_{ñ}$具有相同的多项式阶数 为此，我们提出了一种基于iid随机点的显式（加权最小二乘）算法，并证明了该算法具有正概率。这意味着存在良好的确定性采样算法。

在这里，我们提出了对证明的修改，表明相同的算法至少以概率工作 $\mathrm{1个}-{ñ}^{-C}$ 对于任何给定 $C>0$。