On alternative quantization for doubly weighted approximation and integration over unbounded domains
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It is known that for a ρ-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_L_1^αf^(r)ψ_L_pn^-r+(1/p-1/q)_+, where ω=ρ/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 ρ-weighted integration over unbounded domains.
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