On alternative quantization for doubly weighted approximation and integration over unbounded domains

It is known that for a ϱ-weighted L_q approximation of single variable functions defined on a finite or infinite interval, whose rth derivatives are in a ψ-weighted L_p space, the minimal error of approximations that use n samples of f is proportional to ‖ω^1∕α‖_L1 ^α‖f^(r)ψ‖_Lp n^{−r+(1∕p−1∕q)₊} , where ω=ϱ∕ψ and α=r−1∕p+1∕q, provided that ‖ω^1∕α‖_L1 <+∞. Moreover, the optimal sample points are determined by quantiles of ω^1∕α. In this paper, we show how the error of the best approximation changes when the sample points are determined by a quantizer κ other than ω. Our results can be applied in situations when an alternative quantizer has to be used because ω is not known exactly or is too complicated to handle computationally. The results for q=1 are also applicable to ϱ-weighted integration over finite and infinite intervals.