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

Abstract

It is known that for a -weighted L q -approximation of single variable functions f with the rth derivatives in a ψ-weighted L p space, the minimal error of approximations that use n samples of f is proportional to ω 1/α α L 1 f (r) ψ Lp n −r+(1/p−1/q) + , where ω = /ψ and α = r − 1/p + 1/q. Moreover, the optimal sample points are determined by quantiles of ω 1/α . In this paper, we show how the error of best approximations 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 -weighted integration over unbounded domains.