## Abstract

It is known that for a ϱ-weighted L_{q} approximation of single variable functions defined on a finite or infinite interval, whose rth 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}‖_{L1 } ^{α}‖f^{(r)}ψ‖_{Lp }n^{−r+(1∕p−1∕q)+ }, where ω=ϱ∕ψ and α=r−1∕p+1∕q, provided that ‖ω^{1∕α}‖_{L1 }<+∞. 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.

Original language | English |
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Article number | 105433 |

Journal | Journal of Approximation Theory |

Volume | 256 |

DOIs | |

State | Published - Aug 2020 |

### Bibliographical note

Publisher Copyright:© 2020 Elsevier Inc.

## Keywords

- Quantization
- piecewise Taylor approximation
- unbounded domains
- weighted approximation
- weighted integration

## ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- General Mathematics
- Applied Mathematics