It is known that for a $\varrho$-weighted $L_q$ approximation of single variable functions defined on a finite or infinite interval, whose $r$th derivatives are in a $\psi$-weighted $L_p$ space, the minimal error of approximations that use $n$ samples of $f$ is proportional to ${\Vert {\omega }^{1∕\alpha }\Vert }_{L_1}^{\alpha }{\Vert f^{\left(r\right)}\psi \Vert }_{L_p}{n}^{-r+{(1∕p-1∕q)}_{+}},$ where $\omega =\varrho ∕\psi$ and $\alpha =r-1∕p+1∕q,$ provided that ${\Vert {\omega }^{1∕\alpha }\Vert }_{L_1}<+\infty .$ Moreover, the optimal sample points are determined by quantiles of ${\omega }^{1∕\alpha }.$ In this paper, we show how the error of the best approximation changes when the sample points are determined by a quantizer $\kappa$ other than $\omega .$ Our results can be applied in situations when an alternative quantizer has to be used because $\omega$ is not known exactly or is too complicated to handle computationally. The results for $q=1$ are also applicable to $\varrho$-weighted integration over finite and infinite intervals.

众所周知，对于 $\varrho$加权 ${\mathrm{大号}}_q$ 在有限或无限区间上定义的单变量函数的逼近，其 $\mathrm{[R}$导数在 $\psi$加权 ${\mathrm{大号}}_p$ 空间，使用的近似值的最小误差 $ñ$ 的样本 $F$ 与...成正比 ${\Vert {\omega }^{\mathrm{1个}∕\alpha }\Vert }_{{\mathrm{大号}}_{\mathrm{1个}}}^{\alpha }{\Vert F^{（\mathrm{[R}）}\psi \Vert }_{{\mathrm{大号}}_p}{ñ}^{-\mathrm{[R}+{（\mathrm{1个}∕p-\mathrm{1个}∕q）}_{+}}，$ 哪里 $\omega =\varrho ∕\psi$ 和 $\alpha =\mathrm{[R}-\mathrm{1个}∕p+\mathrm{1个}∕q，$ 规定 ${\Vert {\omega }^{\mathrm{1个}∕\alpha }\Vert }_{{\mathrm{大号}}_{\mathrm{1个}}}<+\infty 。$ 而且，最佳采样点由 ${\omega }^{\mathrm{1个}∕\alpha }。$ 在本文中，我们展示了当量化点确定采样点时最佳近似误差如何变化 $\kappa$ 以外 $\omega 。$ 我们的结果可以应用在必须使用替代量化器的情况下，因为 $\omega$尚不完全清楚或过于复杂以至于无法进行计算处理。的结果$q=\mathrm{1个}$ 也适用于 $\varrho$有限和无限区间内的加权积分。