莫朗集上加倍测度量子误差的渐近均匀性

朱三国

数学学报 ›› 2021, Vol. 64 ›› Issue (5) : 821-838.

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数学学报 ›› 2021, Vol. 64 ›› Issue (5) : 821-838. DOI: 10.12386/A2021sxxb0069
论文

莫朗集上加倍测度量子误差的渐近均匀性

    朱三国
作者信息 +

The Asymptotic Uniformity of the Quantization Error for Doubling Measures on Moran Sets

    San Guo ZHU
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摘要

我们研究了莫朗集E上的加倍概率测度μ的量子误差的渐近性质.对μ的任一rn-最优集αnαn对应的任一Voronoi分划{Paαn)}aαn,定义Iaαnμ)=Paαndxarx);Jαnμ):=minaαnIaαnμ),Jαnμ):=maxaαnIaαnμ).记en,rμ)为测度μrn-级量子误差.在一定意义的开集条件下,我们对加倍测度μ证明了Gersho猜测的下述弱形式:Jαnμ),Jαnμ)≍1/nen,rrμ).

Abstract

We study the quantization errors for the doubling probability measures μ which are supported on Moran sets E ⊂ Rq. For each n ≥ 1, let αn be an arbitrary n-optimal set for μ of order r and {Pa(αn)}aαn an arbitrary Voronoi partition with respect to αn. Let Ia(αn, μ):= Pa(αn) d(x, a)r(x). We define J(αn, μ):=min aαn Ia(αn, μ), J(αn, μ):=maxaαn Ia(αn, μ). Let en,r(μ) denote the nth quantization error for μ of order r. Assuming a version of the open set condition for E, we show that both J(αn, μ) and J(αn, μ) are of the same order as n/1 en,rr (μ).

关键词

莫朗集 / 加倍测度 / 量子误差 / 渐近均匀性 / 开集条件

Key words

Moran sets / doubling measures / quantization error / asymptotic uniformity / open set condition

引用本文

导出引用
朱三国. 莫朗集上加倍测度量子误差的渐近均匀性. 数学学报, 2021, 64(5): 821-838 https://doi.org/10.12386/A2021sxxb0069
San Guo ZHU. The Asymptotic Uniformity of the Quantization Error for Doubling Measures on Moran Sets. Acta Mathematica Sinica, Chinese Series, 2021, 64(5): 821-838 https://doi.org/10.12386/A2021sxxb0069

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基金

国家自然科学基金资助项目(11571144)
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