A characterization of the optimal sets for self-similar measures with respect to the geometric mean error

作者：Zhu S^*

来源：Acta Mathematica Hungarica, 2013, 138(3): 201-225.

DOI：10.1007/s10474-012-0293-5

• self-similar measure
• quantization
• geometric mean error
• Voronoi partition
• logarithmic difference

Let mu be a self-similar measure on a"e (q) associated with a set of contractive similitudes and a probability vector . Assume that satisfies the strong separation condition. In terms of the n-optimal sets for a finite number naa"center dot, we give a characterization for the n-optimal sets for all naa"center dot in the quantization for mu 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 mu-measure of every element of an arbitrary Voronoi partition with respect to an n-optimal set is uniformly comparable with n (-1), provided that mu vanishes on every hyperplane in a"e (q) .

• 出版日期2013-2
• 单位江苏理工学院