Abstract
Let μ be a self-similar measure on ℝq associated with a set of contractive similitudes \({\{S_{i}\}}_{i=1}^{N}\) and a probability vector \({(p_{i})}_{i=1}^{N}\). Assume that \({\{S_{i}\}}_{i=1}^{N}\) satisfies the strong separation condition. In terms of the n-optimal sets for a finite number n∈ℕ, we give a characterization for the n-optimal sets for all n∈ℕ in the quantization for μ with respect to the geometric mean error. As an application, we determine the convergence order for the logarithmic difference of the asymptotic geometric mean error. This characterization also allows us to show that the μ-measure of every element of an arbitrary Voronoi partition with respect to an n-optimal set is uniformly comparable with n −1, provided that μ vanishes on every hyperplane in ℝq.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. A. Bucklew and G. L. Wise, Multidimensional asymptotic quantization with rth power distortion measures, IEEE Trans. Inform. Theory, 28 (1982), 239–247.
I. Csiszár, Generalized entropy and quantization problems, in: Trans. 6th Prague Conf. Inform. Theory, Statist. Decision Functions, Random Processes (1973), pp. 159–174.
P. Elias, Bounds and asymptotes for the performance of multi-variate quantizers, Ann. Math. Stat., 41 (1970), 1249–1259.
A. Gersho, Asymptotically optimal block quantization, IEEE Trans. Inform. Theory, 25 (1979), 373–380.
S. Graf and H. Luschgy, Foundations of Quantization for Probability Dributions, Lecture Notes in Math. 1730, Springer-Verlag (2000).
S. Graf and H. Luschgy, Asymptotics of the quantization error for self-similar probabilities, Real. Anal. Exchange, 26 (2000/2001), 795–810.
S. Graf and H. Luschgy, The quantization dimension of self-similar probabilities, Math. Nachr., 241 (2002), 103–109.
S. Graf and H. Luschgy, Quantization for probabilitiy measures with respect to geometric mean error, Math. Proc. Camb. Phil. Soc., 136 (2004), 687–717.
S. Graf, H. Luschgy and G. Pagès, Distortion mismatch in the quantization of probability measures, ESAIM; Probability and Statistics, 12 (2008), 127–153.
R. Gray and D. Neuhoff, Quantization IEEE Trans. Inform. Theory, 44 (1998), 2325–2383.
P. M. Gruber, Optimum quantization and its applications, Adv. Math., 186 (2004), 456–497.
J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713–747.
L. J. Lindsay and R. D. Mauldin, Quantization dimension for conformal iterated function systems, Nonlinearity, 15 (2002), 189–199.
G. Pagès, A space quantization method for numerical integration, J. Comput. Appl. Math., 89 (1997), 1–38.
K. Pötzelberger, The quantization dimension of distributions, Math. Proc. Camb. Phil. Soc., 131 (2001), 507–519.
P. L. Zador, Development and evaluation of procedures for quantizing multivariate dributions, PhD Thesis, Stanford University, 1964.
S. Zhu, Asymptotic uniformity of the quantization error of self-similar measures, Math. Z., 267 (2011), 915–929.
S. Zhu, The quantization for self-conformal measures with respect to the geometric mean error, Nonlinearity, 23 (2010), 2849–2876.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author is supported by the Project-sponsored by SRF for ROCS, SEM and NNSF of China (11071090).
Rights and permissions
About this article
Cite this article
Zhu, S. A characterization of the optimal sets for self-similar measures with respect to the geometric mean error. Acta Math Hung 138, 201–225 (2013). https://doi.org/10.1007/s10474-012-0293-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-012-0293-5