Asymptotic uniformity of the quantization error of self-similar measures

作者：Zhu, Sanguo^*

来源：Mathematische Zeitschrift, 2011, 267(3-4): 915-929.

DOI：10.1007/s00209-009-0653-1

• Self-similar measure
• Quantization error
• Asymptotic uniformity
• Voronoi partition
• Finite maximal anti-chain

Let mu be a self-similar measure on R(d) associated with a family of contractive similitudes {S(1), ....., S(N)} and a probability vector {p(1,) ...., PN}. Let (alpha(n))(infinity)(n=1) be a sequence of n-optimal sets for mu of order r. For each n, we denote by {P(a) (alpha(n)): a is an element of alpha(n)} a Voronoi partition of R(d) with respect to alpha(n). Under the strong separation condition for {S(1,) ....., S(N)}, we show that the nth quantization error of mu of order r is an element of [1, infinity) satisfies the following asymptotic uniformity proerty: integral(Pa(alpha n)) d(chi, a)(r) d mu(chi) asymptotic to 1/n V(n),(r)(mu), for all a is an element of alpha(n.)

• 出版日期2011-4
• 单位华中科技大学