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 r th 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 ‖ L 1 α ‖ f (r) ψ ‖ L p n − r + (1 ∕ p − 1 ∕ q) + , where ω = ϱ ∕ ψ and α = r − 1 ∕ p + 1 ∕ q , provided that ‖ ω 1 ∕ α ‖ L 1 < + ∞. 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. [ABSTRACT FROM AUTHOR]