On alternative quantization for doubly weighted approximation and integration over unbounded domains
It is known that for a $\rho$-weighted $L_q$-approximation of single variable functions $f$ with the $r$th derivatives in a $\psi$-weighted $L_p$ space, the minimal error of approximations that use $n$ samples of $f$ is proportional to $\|\omega^{1/\alpha}\|_{L_1}^\alpha\|f^{(r)}\psi\|_{L_p}n^{-r+(1/p-1/q)_+},$ where $\omega=\rho/\psi$ and $\alpha=r-1/p+1/q.$ Moreover, the optimal sample points are determined by quantiles of $\omega^{1/\alpha}.$ In this paper, we show how the error of best approximations 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 $\rho$-weighted integration over unbounded domains.