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

TL;DR

This paper investigates how alternative quantization methods affect the accuracy of weighted function approximation and integration over unbounded domains, especially when the optimal quantizer is unknown or complex.

Contribution

It extends existing approximation error analysis to cases using different quantizers, providing practical insights for situations with uncertain or complicated weight functions.

Findings

Error bounds are derived for alternative quantizers in weighted approximation.

Results apply to weighted integration over unbounded domains when q=1.

The analysis helps in choosing quantizers when the optimal is unknown or complex.

Abstract

It is known that for a $\rho$-weighted $L_q$-approximation of single variable functions $f$ with the $r$th derivatives in a $\psi$-weighted $L_p$ space, the minimal error of approximations that use $n$ samples of $f$ is proportional to $\|\omega^{1/\alpha}\|_{L_1}^\alpha\|f^{(r)}\psi\|_{L_p}n^{-r+(1/p-1/q)_+},$ where $\omega=\rho/\psi$ and $\alpha=r-1/p+1/q.$ Moreover, the optimal sample points are determined by quantiles of $\omega^{1/\alpha}.$ In this paper, we show how the error of best approximations changes when the sample points are determined by a quantizer $\kappa$ other than $\omega.$ Our results can be applied in situations when an alternative quantizer has to be used because $\omega$ is not known exactly or is too complicated to handle computationally. The results for $q=1$ are also applicable to $\rho$-weighted integration over unbounded domains.