Asymptotic uniformity of the quantization error for the Ahlfors-David probability measures

• 作者: Sanguo Zhu
• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2020,（6）
• 页码范围: 1039-1056
• 页数: 18页
• 语种: 英文

摘要

Let μ be an Ahlfors-David probability measure on Rq;therefore,there exist some constants so ＞ 0 and ε0,C1,C2 ＞ 0 such that C1εs0 ≤ μ(B(x,ε)) ≤ C2εs0 for all ε ∈ (0,ε0) and x ∈ supp(μ).For n ≥ 1,let αn be an n-optimal set for μ of order r;furthermore,let {Pa(αn)}a∈αn be an arbitrary Voronoi partition with respect to αn.The n-th quantization error en,r(μ) for μ of order r can be defined as en,r(μ) :=f d(x,αn)rdμ(x).We define Ia(αn,μ) :=fpa(αn) d(x,αn)rdμ(x),a ∈ αn,and prove that,the three quantities (J)(αn,μ) :=mina∈αn Ia(αn,μ),(J)(αn,μ) :=maxa∈αnIa(αn,μ),ern,r(μ)-er+1,r(μ)are of the same order as that of 1/n ern,r(μ).Thus,our result exhibits that,a weak version of Gersho's conjecture holds true for the Ahlfors-David probability measures on Rq.