A note on the quantization for probability measures with respect to the geometric mean error
type: Item
instance of: scholarly article
title: A note on the quantization for probability measures with respect to the geometric mean error
zbMATH Open document ID: 1270.28010
author: Sanguo Zhu
published in: Monatshefte für Mathematik
publication date: 2012-11-21T00:00:00Z
review text: The author studied the quantization with respect to the geometric mean error for a probability measure \(\mu\) on \(\mathbb{R}^d\) for which there exist some constants \(C,\;\eta >0\) such that \(\mu(B(x,\epsilon))\leq C\epsilon ^{\eta}\) for all \(\epsilon >0\) and all \(x\in \mathbb{R}^d\). For such measure \(\mu\), the author proved that the upper quantization dimension of \(\mu\) is bounded from above by its upper packing dimension and the lower quantization of \(\mu\) 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 Sierpinski carpets and self-conformal measures. Moreover, based on his previous work, the author proved that the upper and lower quantization coefficient for a self-conformal measure are both positive and finite.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 94A15
zbMATH DE Number: 6109331
zbMATH Keywords: quantization
zbMATH Keywords: geometric mean error
zbMATH Keywords: self-affine measures
zbMATH Keywords: general Sierpiński carpets
zbMATH Keywords: self-conformal measures
MaRDI profile type: Publication
full work available at URL: s00605-011-0307-3
OpenAlex ID: W1966992751
cites work: Bounds and Asymptotes for the Performance of Multivariate Quantizers
cites work: Techniques in fractal geometry
cites work: Foundations of quantization for probability distributions
cites work: Asymptotics of the quantization errors for self-similar probabilities
cites work: Quantization for probability measures with respect to the geometric mean error
cites work: Multiscale self-affine Sierpinski carpets
cites work: Q3731958
cites work: The singularity spectrum for general Sierpiński carpets
cites work: Q4024265
cites work: Quantization dimension for conformal iterated function systems
cites work: The Hausdorff dimension of general Sierpiński carpets
cites work: A space quantization method for numerical integration
cites work: Self-conformal multifractal measures
cites work: Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets
cites work: The quantization dimension of distributions
cites work: Asymptotic quantization error of continuous signals and the quantization dimension
cites work: The quantization for self-conformal measures with respect to the geometric mean error
DOI: 10.1007/S00605-011-0307-3
Recommended article: Quantization for probability measures with respect to the geometric mean error
Recommended article: Asymptotics of the geometric mean error in the quantization for in-homogeneous self-similar measures
Recommended article: Asymptotic of the geometric mean error in the quantization of recurrent self-similar measures
Recommended article: The quantization for self-conformal measures with respect to the geometric mean error
Recommended article: Asymptotics of the geometric mean error in the quantization for product measures on Moran sets
Recommended article: Asymptotic uniformity of the quantization error for the Ahlfors-David probability measures
Recommended article: Asymptotics of the quantization errors for self-similar probabilities
Recommended article: Quantization dimension for random self-conformal measures with respect to the geometric error
Recommended article: THE ESTIMATE OF THE ERROR FOR APPROXIMATION OF SOME DISCRETE DISTRIBUTION BY THE GEOMETRIC DISTRIBUTION
Recommended article: Asymptotic order of the quantization errors for a class of self-affine measures
description: scientific article; zbMATH DE number 6109331