Monatshefte für Mathematik

, Volume 167, Issue 2, pp 291–305 | Cite as

A note on the quantization for probability measures with respect to the geometric mean error

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Abstract

We study the quantization with respect to the geometric mean error for probability measures μ on \({\mathbb{R}^d}\) for which there exist some constants C, η > 0 such that \({\mu(B(x,\varepsilon))\leq C\varepsilon^\eta}\) for all ε > 0 and all \({x\in\mathbb{R}^d}\) . For such measures μ, we prove that the upper quantization dimension \({\overline{D}(\mu)}\) of μ is bounded from above by its upper packing dimension and the lower one \({\underline{D}(\mu)}\) is bounded from below by its lower Hausdorff dimension. This enables us to calculate the quantization dimension for a large class of probability measures which have nice local behavior, including the self-affine measures on general Sierpiński carpets and self-conformal measures. Moreover, based on our previous work, we prove that the upper and lower quantization coefficient for a self-conformal measure are both positive and finite.

Keywords

Quantization Geometric mean error Self-affine measures General Sierpiński carpets Self-conformal measures 

Mathematics Subject Classification (2000)

28A80 28A78 94A15 
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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhanChina

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