caa a22 4500 445836083 CHVBK 20180317145330.0 cr unu---uuuuu 170323e20110401xx s 000 0 eng 10.1007/s00209-009-0653-1 doi (NATIONALLICENCE)springer-10.1007/s00209-009-0653-1 Zhu Sanguo Department of Mathematics, Huazhong University of Science and Technology, 430074, Wuhan, China aut Asymptotic uniformity of the quantization error of self-similar measures [Elektronische Daten] [Sanguo Zhu] Letμ be a self-similar measure on $${\mathbb{R}^d}$$ associated with a family of contractive similitudes {S 1, . . . , S N} and a probability vector {p 1, . . . , p N}. Let $${(\alpha_n)_{n=1}^\infty}$$ be a sequence of n-optimal sets forμ of order r. For each n, we denote by $${\{P_a(\alpha_n) : a \in \alpha_n\}}$$ a Voronoi partition of $${\mathbb{R}^d}$$ with respect to α n. Under the strong separation condition for {S 1, . . . , S N}, we show that the nth quantization error ofμ of order $${r \in [1, \infty)}$$ satisfies the following asymptotic uniformity property: $$\int \limits _{P_a(\alpha_n)}{\rm d}(x, a)^rd\mu(x) \asymp \frac{1}{n}V_{n,r}(\mu),\quad {\rm for\,all}\,a \in \alpha_n.$$ Springer-Verlag, 2009 Self-similar measure nationallicence Quantization error nationallicence Asymptotic uniformity nationallicence Voronoi partition nationallicence Finite maximal anti-chain nationallicence Mathematische Zeitschrift Springer-Verlag 267/3-4(2011-04-01), 915-929 0025-5874 267:3-4<915 2011 267 209 https://doi.org/10.1007/s00209-009-0653-1 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00209-009-0653-1 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Zhu Sanguo Department of Mathematics, Huazhong University of Science and Technology, 430074, Wuhan, China aut NATIONALLICENCE 773 0- Mathematische Zeitschrift Springer-Verlag 267/3-4(2011-04-01), 915-929 0025-5874 267:3-4<915 2011 267 209 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer