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Asymptotic order of the quantization errors for a class of self-affine measures


Author: Sanguo Zhu
Journal: Proc. Amer. Math. Soc. 146 (2018), 637-651
MSC (2010): Primary 28A80, 28A78; Secondary 94A15
DOI: https://doi.org/10.1090/proc/13756
Published electronically: September 7, 2017
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Abstract: Let $ E$ be a Bedford-McMullen carpet determined by a set of affine mappings $ (f_{ij})_{(i,j)\in G}$ and $ \mu $ a self-affine measure on $ E$ associated with a probability vector $ (p_{ij})_{(i,j)\in G}$. We prove that, for every $ r\in (0,\infty )$, the upper and lower quantization coefficient are always positive and finite in its exact quantization dimension $ s_r$. As a consequence, the $ n$th quantization error for $ \mu $ of order $ r$ is of the same order as $ n^{-\frac {1}{s_r}}$. In sharp contrast to the Hausdorff measure for Bedford-McMullen carpets, our result is independent of the horizontal fibres of the carpets.


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Additional Information

Sanguo Zhu
Affiliation: School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, People’s Republic of China
Email: sgzhu@jsut.edu.cn

DOI: https://doi.org/10.1090/proc/13756
Keywords: Quantization error, convergence order, quantization coefficient, self-affine measures, Bedford-McMullen carpets.
Received by editor(s): December 21, 2016
Received by editor(s) in revised form: March 22, 2017
Published electronically: September 7, 2017
Additional Notes: The author was supported by NSFC 11571144
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2017 American Mathematical Society

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