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

Abstract

It is known that for a ϱ-weighted Lq approximation of single variable functions defined on a finite or infinite interval, whose rth derivatives are in a ψ-weighted Lp space, the minimal error of approximations that use n samples of f is proportional to ‖ω1∕α‖L1α‖f(r)ψ‖Lpn−r+(1∕p−1∕q)+, where ω=ϱ∕ψ and α=r−1∕p+1∕q, provided that ‖ω1∕α‖L1<+∞. Moreover, the optimal sample points are determined by quantiles of ω1∕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.