Asymptotics of the geometric mean error in the quantization for in-homogeneous self-similar measures

作者：Zhu, Sanguo^*; Zhou, Youming; Sheng, Yongjian

来源：Journal of Mathematical Analysis and Applications, 2016, 434(2): 1394-1418.

DOI：10.1016/j.jmaa.2015.09.081

• Geometric mean error
• In-homogeneous self-similar measures
• Open set condition
• Quantization coefficient

Let (f(i))(i=1)(N), be a family of contractive similitudes on R-q satisfying the open set condition. Let (p(i))(i=0)(N) be a probability vector with p(i) > 0 for all i = 0, 1, ..., N. We study the asymptotic geometric mean errors e(n),0(mu), n >= 1, in the quantization for the in-homogeneous self-similar measure mu, associated with the condensation system ((f(i))(i=1)(N), (p(i))(i=0)(N), nu) We focus on the following two independent cases: (I) nu is a self -similar measure on R-q associated with (f(i))(i=1)(N) ; (II) nu is a self -similar measure associated with another family of contractive similitudes (g(i))(i=1)(M), on R-q satisfying the open set condition and ((f(i))(i=1)(N), (p(i))(i=0)(N), nu) satisfies a version of in-homogeneous open set condition. We show that, in both cases, the quantization dimension D-0(mu) of mu, of order zero exists and agrees with that of nu, which is independent of the probability vector (p(i))(i=0)(N). We determine the exact convergence order of (e(n),0(mu))(n=1)(infinity); namely, for D-0(mu) =: d(0), there exists a constant D > 0, such that
D-1n (-1/d0) <= e(n,0)(mu) <= Dn(-1/d0), n >= 1.

• 出版日期2016-2-15
• 单位江苏大学